WASHINGTON — As President Obama and Mitt Romney rush to define the other, each man is alleging one of the great economic sins: that his rival accelerated the exodus of American jobs to foreign countries.

Mr. Obama accuses Mr. Romney of being at the helm of a firm that invested in companies that outsourced their jobs. Mr. Romney, in remarks on Tuesday, called Mr. Obama the real “outsourcer in chief” for sending billions of dollars in stimulus funds to foreign-based firms and companies that “end up making their products outside the United States.”

The particulars have been denied or defended by both sides, and in many cases picked over by independent fact-checkers. In response, the two candidates and their allies have all but stuck their fingers in their ears while continuing with their outsourcing attacks.

There is no question of the potential power of the narrative, coming at a time of economic uncertainty and frustration about the tepid expansion of the labor force. But in both cases, there are deep questions about the fundamental truth of the arguments, even as each campaign continues to press forward with its charges. Mr. Obama’s attacks on Mr. Romney’s time at Bain Capital rely on thinly documented examples of outsourcing that took place when the Republican candidate was, at best, an absentee owner of his firm, having left day-to-day management to run the 2002 Winter Olympics in Salt Lake City.