When President Biden agreed to bipartisan talks on border legislation last fall, Democratic strategists hoped a deal might take the issue off the table for his re-election campaign.

But with the collapse of the resulting bipartisan immigration agreement this week at the hands of former President Donald J. Trump, Mr. Biden got something else instead: someone to blame.

The crisis at the southwestern border has been one of the most vexing challenges of Mr. Biden’s presidency, one that has defied his policy prescriptions and drained his public support. With record numbers of migrants illegally crossing into the country, the president has come under pressure from Democrats as well as Republicans to take more action.

For three years, Mr. Biden struggled to offer voters a compelling answer to the question of why the border has turned into such a crisis on his watch. He has avoided public discussion of the issue as much as possible, preferring to focus his messaging on other priorities. But with Mr. Trump’s intervention persuading congressional Republicans to abandon the border deal that they themselves had demanded, Mr. Biden finally has an opportunity to shift from defense to offense.

“The American people are going to know why it failed,” he declared in a televised speech at the White House. “I’ll be taking this issue to the country, and the voters are going to know that it’s not just a moment — just at the moment we were going to secure the border and fund these other programs, Trump and the MAGA Republicans said no because they’re afraid of Donald Trump.”

“Every day between now and November,” he added, “the American people are going to know that the only reason the border is not secure is Donald Trump and his MAGA Republican friends.”

Mr. Trump and his allies ridiculed the idea that Mr. Biden could deflect blame after three years of failing to secure the border.

“Joe Biden blamed President Trump for the border crisis that Biden himself created,” said Karoline Leavitt, a spokeswoman for the former president. “This is a brazen, pathetic lie and the American people know the truth — President Trump’s policies created the most secure border in American history, and it was Joe Biden who reversed them.”

Even as Republicans took their cue from the former president and rejected the deal as inadequate, they tried to make their point on the Biden administration’s failure on immigration by impeaching Alejandro N. Mayorkas, the secretary of homeland security. But the bid fell a vote short on the House floor on Tuesday, an embarrassing setback for the G.O.P., which now has to decide whether to try again in the weeks to come.

The border has been one of Mr. Biden’s least favorite issues. Illegal crossings have shot up since he took office, from 73,944 reported in December 2020 just before he was inaugurated to 302,034 last December, and governors and mayors as far away as New York and Illinois have sounded alarms about the resulting burdens on their communities.

Forty-five percent of Americans now view the situation at the border as “a crisis,” up 8 percentage points from last spring, and another 30 percent consider it “very serious,” according to a poll by CBS News and YouGov last month. A survey released Wednesday by PBS NewsHour, NPR and Marist found that only 29 percent of Americans approve of Mr. Biden’s leadership on the issue, as more Democrats and independents express concern.

As a matter of pure politics, Mr. Biden was probably never going to outperform his challenger among voters who care strongly about illegal immigration, Mr. Trump’s signature issue since the days he led crowds chanting “build the wall” in 2016.

But in terms of re-election strategy, Democratic operatives believed that Mr. Biden needed to keep immigration from cutting into his support among swing voters disturbed by the surge of undocumented migrants without alienating progressives who have been disappointed that he has not done more to reverse Trump-era policies.

It was a measure of how much the politics of the issue have shifted in recent years that Mr. Biden embraced the bipartisan deal negotiated by Senators James Lankford, Republican of Oklahoma; Christopher S. Murphy, Democrat of Connecticut; and Kyrsten Sinema, the Democrat-turned-independent of Arizona.

The legislation would have tightened the rules for asylum seekers, expanded detention facilities, hired more border agents, sped up the process to send back migrants who do not qualify and even shut down the border temporarily during peak times. But it incorporated none of the signature provisions long demanded by Democrats in comprehensive immigration legislation, such as a pathway to citizenship for those already here or protections for younger immigrants brought into the country as children.

Mr. Trump made clear that he saw the deal not as a solution but a threat to his bid to reclaim his office. “This Bill is a great gift to the Democrats, and a Death Wish for The Republican Party,” he wrote on social media this week. “It takes the HORRIBLE JOB the Democrats have done on Immigration and the Border, absolves them, and puts it all squarely on the shoulders of Republicans. Don’t be STUPID!!!”

The White House wasted little time reframing the issue as an obstructionist Mr. Trump intimidating Republicans into turning on a deal that has the support of conservative institutions, including the Border Patrol union that has previously endorsed Mr. Trump. “Will the House G.O.P. vote with the Border Patrol to secure the border, or with Donald Trump for more fentanyl?” the White House asked in a memo sent to reporters.

The change was welcome for Democrats looking ahead to a close election.

“Until very recently, the border was President Biden’s problem almost exclusively,” said Geoffrey Garin, a Democratic pollster. “But now by blocking strong, bipartisan border legislation Republicans have made it their problem as well.”

He added, “The fact that President Biden can now say that he was prepared to sign and enforce the strongest border law in history, but Republicans blocked it at Trump’s behest puts Biden in a much better position than he was before in the immigration debate.”

Margie Omero, another Democratic strategist, said voters would understand which side actually wanted to get something done. “Joe Biden and Democrats in Congress are working for solutions,” she said. “The Republican Party routinely puts obstruction and scoring political points over tackling our big challenges.”

Mr. Biden’s critics, though, doubt he can shift blame after so much time. For much of his presidency, they said, the president and his allies have resisted even admitting there was a crisis, only to switch gears and say that there is one and that it is Mr. Trump’s fault.

“It seems to me absurd on its face,” said Mark S. Krikorian, the executive director of the Center for Immigration Studies and a leading voice for tougher policies. “Obviously Biden partisans will latch onto that, and obviously Trump partisans will scoff at it. The question is will people in the middle buy it or not. I find it hard to believe that anybody would believe it. After three years?”

Scott Jennings, a Republican strategist, called it “a transparently cynical ploy” that will not work. “He must really think voters are stupid, trying to convince them that after three years of his policies that Republicans are somehow at fault,” Mr. Jennings said. “Nobody believes Joe Biden wants to ‘get tough’ on the border. Please. His administration has argued for three years the border is secure. What changed? Oh. It’s election time.”

Elections, of course, revolve around narratives. For three years, Republicans had a clear story line when it came to the border — Mr. Biden either intentionally or incompetently opened the floodgates. Now the president has a counternarrative to offer — that whatever may have happened before, at least he wanted to fix the problem and Mr. Trump did not. The next nine months will test which one is more persuasive.