For weeks, President Obama’s top campaign strategists have tried to establish that Mitt Romney was at the helm of Bain Capital when some of its companies moved workers overseas.

Proving his presence at Bain between 1999 and 2002 would create a direct link to what Mr. Obama’s team describes as the outsourcing of American jobs. And it would contradict Mr. Romney’s repeated assertions that by then he had left Bain, the private equity firm he founded, to rescue the Salt Lake City Olympic Games.

But definitive proof about Mr. Romney’s activities remains elusive, leading to another day of increasingly bitter charges and countercharges from the two campaigns.

On Thursday — even as Mr. Romney started broadcasting a television commercial accusing the president of approving dishonest attacks — Mr. Obama’s campaign seized on a new report in The Boston Globe about Securities and Exchange Commission documents showing that Mr. Romney was listed as the chief executive, president and owner of Bain Capital during that period.