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“One more thing: What if I told you the man I’m talking about isn’t him, it’s him,” she says, as the backdrop morphs from a photo of Mr. Obama to a solemn-faced Mr. Romney.

Andrea Saul, a Romney spokeswoman, countered that Mr. Santorum’s claims are false and pointed to independent fact-checkers who came to similar conclusions.

Rick Santorum is attacking pollsters, attacking reporters and attacking Mitt Romney,” she said, alluding to Mr. Santorum’s recent public spat with a New York Times reporter, which he has since used to try to raise money.

“It is sad to see him completely lose his bearings and revert to patently false claims. Sen. Santorum is at a point of desperation that he will say or do anything. It is pretty clear that he is lashing out at everyone around him in order to prop up his sinking campaign,” Ms. Saul said.

The battle Tuesday follows a USA Today/Gallup poll that found Mr. Obama holds an eight-point edge over Mr. Santorum and a four-point edge over Mr. Romney among registered voters nationally, giving the Democrat his largest lead over Mr. Romney since the group started asking the question in August.

Perhaps more important, the survey showed that Mr. Obama is outperforming both Republican men in a dozen swing states, especially among women.

“The momentum in the 2012 U.S. presidential election appears to be going in Obama’s favor; he now enjoys his best positioning against likely Republican nominee Romney nationally and in key swing states to this point in the campaign,” Jeffrey M. Jones of Gallup said.

This article is based in part on wire service reports.