The president’s political strategists were focusing on Florida, already a swing state but now perhaps ground zero for attacks on Mr. Ryan’s plans to restructure Medicare, while monitoring Wisconsin to see if the Republicans tried to invest in Mr. Ryan’s home state. Congressional election strategists were mapping out plans to force every Republican candidate for the House and Senate to own or disown Mr. Ryan’s budget plan.

“This is the perfect choice for us to finish our frame of Romney,” said Eddie Vale, a spokesman for Workers’ Voice, an A.F.L.-C.I.O. “ super PAC.” “What Romney and Bain did to working families and companies is what Romney and Ryan would do to all Americans.”

Bill Burton, one of the founders of Priorities USA Action, a pro-Obama super PAC, said his group had always planned to devote the fall to tying Mr. Romney to Mr. Ryan’s budget plan, but had worried that it might not stick.

“From the beginning when we polled, we found that the Ryan plan was the most toxic political document ever, but the problem was you couldn’t convince voters that any politician would actually support it,” Mr. Burton said. “Now this actually makes the job easier.”

Stanley B. Greenberg, a Democratic pollster, said that surveys showed attacks on the Ryan plan moved voters dramatically, adding that Mr. Romney could no longer dance around it. “There is no longer a nuance to that debate,” he said. “There are a lot of white, working-class voters, particularly older ones, who will walk away now, even if tempted earlier by the slow economy.”

The Romney camp said Democrats were fooling themselves. “The race is now framed exactly as we want it,” said Kevin Madden, a senior Romney adviser. “Voters are going to judge our current struggling economy and President Obama’s lack of leadership on that issue very harshly, and then look at a Romney-Ryan ticket as an opportunity to take the country in a bold new direction towards a better future.”

The looming clash reflects an intriguing evolution from just two years ago, when Mr. Obama expressed a grudging admiration for Mr. Ryan, another young politician with big ideas. Asked in the fall of 2010 which Republicans he could envision working with in the next Congress, Mr. Obama mentioned only one who would still be around, Mr. Ryan.

He said Mr. Ryan was “absolutely sincere about wanting to reduce the deficit,” though he quarreled with his approach. “I give him credit for at least being willing to put out there some tough choices,” Mr. Obama said, “although, as I said, even there, the numbers don’t quite match up the way they should.”

What Mr. Obama found appealing, the notion of a man of ideas willing to make tough choices, is what he now will need to devalue him. While Democrats openly crowed that Mr. Ryan was the choice they had hoped for because of the sharp contrast, they acknowledged he is a young, attractive, well-spoken politician who explains his plan better than his fellow Republicans do. If he makes the issue the national debt that has risen so much under Mr. Obama, rather than his solution for it, Mr. Ryan could pose a serious challenge to the president.

In other words, to put the bull’s-eye on Mr. Ryan’s back, Mr. Obama will first have to get it off his own.