As President Trump looked ahead this week to his high-profile, high-risk meeting with President Vladimir V. Putin of Russia, he reflected momentarily on the curious and confusing relationship between the two men. “I got along well with Putin,” he said.

Intriguingly, he used the past tense. Over the past few weeks, Mr. Trump has expressed rare frustration over the Russian leader’s unwillingness to make peace in Ukraine. But as the president elaborated earlier this week, he sounded eager to switch back to the present tense when the two sit down in Alaska on Friday.

Mr. Trump’s affinity for the iron-fisted master of the Kremlin has perplexed much of the political and diplomatic world for the past decade, challenging assumptions, fueling investigations, reshaping elections and upending alliances. Now the relationship faces its most critical test as Mr. Trump seeks to broker a halt to the war in Ukraine: Is he ready to put serious pressure on his Russian counterpart? Will Mr. Putin again win over the president to his way of thinking? Or is their friendship really on the rocks?

For all of his recent complaints about Russian intransigence and demands that the war stop, Mr. Trump has still largely held back from harsh criticism of Mr. Putin personally, preferring to use words like “disappointed” and “not happy.” He has aimed his sharpest broadsides instead at Dmitri A. Medvedev, the former caretaker president who has engaged in belligerent talk, a way of avoiding going after Mr. Putin more directly.

And in recent days, Mr. Trump has seemed to pivot back to his posture from earlier in the year, when he directed more blame for the war on Ukraine’s president, Volodymyr Zelensky, even though it was Mr. Putin who invaded his neighbor in the first place and has rejected American proposals for an unconditional cease-fire.

“Putin clearly pushed the envelope beyond what Trump was willing to take, and that explains why in the past month or six weeks you’ve seen increasingly negative comments about Putin,” John R. Bolton, a national security adviser to Mr. Trump in his first term and now a vocal critic, said in an interview. “His friend isn’t helping him out here. He’s not getting to a deal.”

But Mr. Bolton said Mr. Trump clearly has not given up on his friend. By inviting Mr. Putin to American soil despite U.S. sanctions and an international arrest warrant for war crimes, he said, Mr. Trump has rewarded the Russian leader, effectively freeing him from the international isolation box established by Mr. Trump’s predecessor, President Joseph R. Biden Jr., and European leaders.

“He just doesn’t know enough to not get played,” Mr. Bolton said of Mr. Trump. “He wants to get along. He thinks he’s friends with Putin. I don’t think Putin thinks he’s friends with him. Putin’s as coldblooded as they come.”

Longtime analysts of the U.S.-Russian relationship said that Mr. Trump’s recent words of exasperation do not signal a real break with Mr. Putin, at least not yet.

“I don’t think there has been any meaningful shift in Trump’s views of Putin or his desire to achieve a reset,” said Angela E. Stent, a national intelligence officer on Russia during President George W. Bush’s administration. “It is true that he did express frustration with Putin and praise for Zelensky a few weeks ago, but that seems to have evaporated once he realized that Putin would not agree to anything before Trump’s imposed deadline” for a cease-fire.

Michael A. McFaul, who was ambassador to Russia under President Barack Obama, said he was struck by Mr. Trump’s momentary shift in tone a few weeks ago. “But lately he has drifted back to his old self,” he said, “again blaming Zelensky in part for Putin’s invasion, not entertaining Zelensky’s proposal for a trilateral meeting, and already suggesting that Zelensky is going to have to make major concessions, but saying nothing about what concessions need to be made by Putin.”

The meeting in Alaska will be Mr. Putin’s first visit to the United States outside of the United Nations since 2007, when Mr. Bush invited him to his family’s compound in Kennebunkport, Maine. After multiple phone calls and forays by his special envoy, Steve Witkoff, Mr. Trump said he wanted to get in the same room with Mr. Putin to break the logjam.

“President Putin invited me to get involved,” Mr. Trump said earlier this week. “He wants to get involved. I think, I believe he wants to get it over with. Now, I’ve said that a few times and I’ve been disappointed because I’d have like a great call with him and then missiles would be lobbed into Kyiv or some other place and you’d have 60 people laying on a road dying.”

But Mr. Trump said he thought he could get through to his Russian counterpart. “I’m going in to speak to Vladimir Putin, and I’m going to be telling him, ‘You got to end this war; you got to end it.’”

Still, his aides have sought to play down expectations of a breakthrough. “This is a listening exercise for the president,” Karoline Leavitt, the White House press secretary, told reporters on Tuesday. “Look, only one party that’s involved in this war is going to be present. And so this is for the president to go and to get, again, a more firm and better understanding of how we can hopefully bring this war to an end.”

Mr. Trump’s admiration for Mr. Putin stretches back years, long before his presidency. After Time magazine named Mr. Putin its person of the year in 2007, Mr. Trump sent a gushing note of congratulations, writing, “I am a big fan of yours!” At the time, Mr. Trump, who long aspired to build a Trump tower in Moscow, was increasingly reliant on Russian money because American banks stopped doing business with him.

When he brought the Miss Universe contest to Moscow in 2013, Mr. Trump flattered Mr. Putin publicly. “Will he become my new best friend?” he wrote on social media. But he gave conflicting accounts of their encounters before his campaign for the White House. After repeatedly saying over the years that he had met Mr. Putin and had “a relationship with him,” Mr. Trump reversed gears during the 2016 campaign after it became politically problematic. “I never met Putin,” he suddenly said. “I don’t know who Putin is.”

Russian intervention in that campaign on Mr. Trump’s behalf proved to be one of the defining issues of his presidency, one that still animates him today as he angrily tries to rewrite the history of that episode and reframe the investigation that followed as a “coup” by Mr. Obama. While Mr. Trump refers to the matter as “the Russia hoax,” investigators documented concerted Russian efforts to tilt the election and unusually extensive contacts between Moscow and people in Mr. Trump’s orbit.

Mr. Trump’s son, son-in-law and campaign manager met during the campaign with a Russian offering dirt on his opponent, Hillary Clinton, what they were told was “part of Russia and its government’s support for Mr. Trump.” His campaign manager passed along internal polling data to a longtime business associate identified as a Russian intelligence operative. When Mr. Trump publicly asked Russia, “if you’re listening,” to hack Mrs. Clinton’s servers, Russian agents did so within hours.

The subsequent investigation by the special counsel Robert S. Mueller III did not establish any illegal coordination between Russia and the campaign and concluded that “the evidence was not sufficient to charge” anyone with criminal conspiracy. Mr. Trump, for his part, said during a 2018 meeting in Helsinki that he accepted Mr. Putin’s denial of election interference over the judgment of American intelligence agencies.

Since reclaiming the White House in January, Mr. Trump has adopted policies welcomed in Moscow. He has all but torn down Voice of America and the National Endowment for Democracy, which have long irritated Russian leaders. He has curbed programs to combat Russian disinformation, election interference, sanctions violations and war crimes.

He has committed no new U.S. military aid to Ukraine beyond that already approved under Mr. Biden. But he agreed to allow European nations to purchase U.S. arms for Ukraine and he has threatened to slap 50 percent tariffs on India, citing its purchase of Russian oil, though he has not similarly penalized China, which buys even more Russian energy.

Many experts, both Republicans and Democrats, have come to the conclusion that Mr. Trump’s desire to forge a close relationship with Mr. Putin is primarily about power and shared authoritarian instincts. Mr. Trump sees himself as a strong man and admires a peer who fits the common definition of a strongman leader.

“I think Putin and Trump recognize that they share a common view of the world — cynical, unsentimental, focused on power, money and territory,” said Daniel Treisman, a Russia scholar at the University of California, Los Angeles. “They feel they understand each other. Putin knows Trump would like to be on his side, but has lately felt ignored. So he’s ready to flatter and distract, with offers to U.S. business and to help in other global arenas.”

Michael C. Kimmage, a historian of U.S.-Russian relations at the Catholic University of America and author of “Collisions,” a book about the Ukraine war, said that Mr. Trump may believe that cultivating Mr. Putin serves a strategic purpose. When Mr. Trump returned to office in January, he said, he “thought he could scale back the war in Ukraine through his personal connection to Putin.”

That did not work, but Mr. Trump may still persist in his personality-driven diplomacy with Mr. Putin because he “cannot resist the media spectacle and the opportunity to place himself at the center of it.” Moreover, after recent interventions to stop fighting between Thailand and Cambodia and settle decades of dispute between Armenia and Azerbaijan, he added, it’s possible that Mr. Trump “thinks he’s on a roll.”

But in pursuing his Alaska summit with Mr. Putin without doing the usual diplomatic spadework of lower-level negotiations and consultations with allies beforehand to ensure success, Mr. Trump is attempting “a new style of diplomatic action,” Mr. Kimmage said. Mr. Trump, he said, appears focused primarily on settling the world’s big issues directly with leaders of the three major powers: himself, Mr. Putin and President Xi Jinping of China.

“This new style will turn much of the world into bystanders and simultaneously elevate a handful of leaders — with Trump first among equals — to a uniquely powerful status, a V.I.P. club of statesmen, consisting primarily of Xi, Putin and Trump,” Mr. Kimmage said. “This impulse will be put to the test on Friday.”