At the beginning of his remarks from the White House on Tuesday announcing that he would prohibit migrants from seeking asylum at the U.S.-Mexico border, President Biden tried to make clear just whose fault it was that he was taking action by executive order.

The White House, Mr. Biden said, had struck an agreement with congressional Republicans earlier this year on what he called the “strongest border security agreement in decades.”

It did not take. Republicans bailed on the deal.

“Why? Because Donald Trump told them to,” Mr. Biden said. “He didn’t want to fix the issue. He wanted to use it to attack me. That’s what he wanted to do.”

On this, Mr. Biden proved correct.

Mr. Trump, the presumptive Republican nominee set to face Mr. Biden in the general election, indeed attacked the president a couple of hours before his border announcement. Mr. Trump, who has made hard-line immigration policies the center of his political identity since the start of his 2016 campaign, derided Mr. Biden’s executive order as too little action taken too late, and he argued that it was timed to benefit the president politically.

“After nearly four years of his failed weak leadership — pathetic leadership — Crooked Joe Biden is pretending to finally do something about the border,” Mr. Trump said in a video posted to his social media site. “But in fact, it’s all about show, because he knows we have a debate coming up in three weeks.”

The former president’s rhetoric echoed much of the reaction from allied Republicans. The Republican National Committee has adopted the alliterative “Biden’s Border Bloodbath.” Representative Richard Hudson of North Carolina, the chairman of the House Republicans’ campaign arm, predicted that voters would be so angry about the border that “in November they will deport House Democrats from their seats for enabling this crisis.”