President Donald Trump threatened New York state lawmaker Zohran Mamdani with arrest if the presumptive Democratic nominee for New York City mayor defies Immigration and Customs Enforcement operations.

The president also continued to allege the 33-year-old Democratic socialist is a "communist" while talking to reporters Tuesday at the new so-called "Alligator Alcatraz" migrant detention center in Florida's Everglades.

When asked by a reporter what his message is to Mamdani -- after he said in a victory speech following the New York City Democratic mayoral primary that he would "stop masked ICE agents from deporting our neighbors" -- Trump responded, "Well then, we'll have to arrest him."

"Look, we don't need a communist in this country, but if we have one, I'm going to be watching over him very carefully on behalf of the nation," Trump continued.

Trump also referenced false claims that Mamdani is in the country illegally.

"A lot of people are saying he's here illegally," Trump said. "We're going to look at everything. Ideally, he's going to turn out to be much less than a communist. But right now he's a communist. That's not a socialist."

Born in Uganda, Mamdani has lived in the United States since he was 7 years old and became a naturalized citizen in 2018.

As he departed the White House earlier Tuesday, Trump continued his attack on Mamdani in the wake of the primary, calling him a "total nut job" and "bad news" and reiterating his false claim the politician is a communist.

"I think I'm gonna have a lot of fun with him, watching him, because he has to come right through this building to get his money," Trump said after threatening to withhold funding from New York if Mamdani doesn't "do the right thing."

When asked about comments from President Donald Trump on social media calling him a "Communist Lunatic" in an interview with ABC News last week, Mamdani shrugged off the epithet, saying he'd encourage Trump to learn about his policies, and that he'd work with Trump on affordability but would resist the president's deportation plans.

"The next mayor of New York City will have to work with the Trump administration. Are you willing to do that? Will you do that?" ABC News Senior Political Correspondent Rachel Scott asked.

"I will work with the Trump administration when it is to the benefit of New Yorkers," Mamdani said. "My approach will never be reflexive, whether in agreement or opposition, but if it comes at the expense of the New Yorkers that I'm running to serve, then, no, I will not be working with the administration on harming the people that I look to represent."

Mamdani also responded to Trump's attacks during an interview on NBC's "Meet the Press" on Sunday. Asked how he responds to being called a communist, Mamdani said, "I am not."

"I have already had to start to get used to the fact that the president will talk about how I look, how I sound, where I'm from, who I am. Ultimately because he wants to distract from what I'm fighting for," Mamdani said. "And I'm fighting for the very working people that he ran a campaign to empower that he has since then betrayed."

The New York City Board of Elections released the results of the ranked-choice voting tabulation on Tuesday, one week after initial results that only reflected voters' first choice candidates showed Mamdani with around 44% of the vote and Cuomo with around 36%.

Mamdani now leads former Gov. Andrew Cuomo by 12 points, 56% to 44%, in the Democratic primary, according to the updated results.

These may not be the final margins, as cured ballots come in before the Board of Elections certifies the election on July 15, though subsequent updates are not expected to change who is on top. Current Mayor Eric Adams will run as an independent, while Cuomo will also run as an independent on the "Fight and Deliver" ballot line, a source close to the campaign confirmed to ABC News. Curtis Sliwa, who also ran for mayor against Adams four years ago, will be the Republican candidate.

The New York City mayoral election is scheduled to occur on Nov. 4.

ABC News' Oren Oppenheim, Brittany Shepherd and Lalee Ibssa contributed to this report.