President Obama’s campaign said Wednesday that Mitt Romney either lied to voters or to federal regulators about how long he ran his company, while the GOP candidate’s team said the “felony” accusation was so over-the-top that the president should apologize for his aides’ behavior; voters were left to sort it all out.

The dust-up over Mr. Romney’s time at private equity firm Bain Capital underscored the growing realization that his business ties will come under scrutiny for the rest of the campaign — a theme the former Massachusetts governor’s rivals started in the Republican primary.

Wednesday’s charges stemmed from a report in the Boston Globe that said documents filed with the Securities and Exchange Commission show Mr. Romney may have led Bain until 2002, three years longer than he claims.

The charge is significant because the Obama campaign says that Mr. Romney shares responsibility for the investments Bain made between 1999 and 2002, which led to jobs being shipped off to China and India.

“Either Mitt Romney, through his own words and his own signature, was misrepresenting his position at Bain to the SEC, which is a felony, or he was misrepresenting his position at Bain to the American people to avoid responsibility for some of the consequences of his investments,” Stephanie Cutter, deputy campaign manager for the Obama campaign, told reporters in a conference call.

“If that’s the case, if he was lying to the American people, then that is a real character and trust issue that the American people need to take very seriously,” she said.

The Romney camp said the Globe report was “not accurate,” and touted the findings of independent fact checkers that agreed with them.

Mr. Romney’s aides also pointed to a story Wednesday by Fortune that said internal Bain documents showed Mr. Romney had stopped managing the firm’s clients when he went to oversee the Salt Lake City Olympics.

In a statement to reporters, Romney campaign manager Matt Rhoades called the Obama team’s accusations “reckless and unsubstantiated.”

“President Obama ought to apologize for the out-of-control behavior of his staff, which demeans the office he holds,” he said.

Bain Capital also released a statement that supported Mr. Romney’s claim that he left in 1999 and that he had “no involvement with the management or investment activities of the firm or with any of its portfolio companies since the day of his departure.”

Democrats have demanded Mr. Romney release more tax returns in order to give voters a better look at his finances, and said Thursday that his role at Bain is critical for voters to have a full picture of how he approaches job creation.

David Yepsen, director of the Paul Simon Public Policy Institute at Southern Illinois, said Mr. Romney’s successful business history will continue to dog him.

“There’s no doubt his complex finances makes this a hard story for Romney to control,” Mr. Yepsen said.

He said the Romney camp deserves blame for not having better answers ready.

“After running for president for years, he and his people should have known this was coming and disclosed a lot more a lot earlier so that this was all an old story by now,” Mr. Yepsen said.

Even before the Globe story took root, the name-calling had begun early in the day when the Romney camp released a new ad, “No Evidence,” that said Mr. Obama was lying to voters in his claim that Mr. Romney had a hand in offshoring jobs.

“When a president doesn’t tell the truth. How Can We Trust Him To Lead?” Mr. Romney’s ad retorts.

The ad also highlighted the analysis of Factcheck.org, which found there was “no evidence” that Mr. Romney shipped jobs overseas, as well as The Washington Post, which had reported the story the Obama campaign cited but whose fact-checker called the charge “misleading.”

Ms. Cutter had a different take.

The Globe report, she said, proved that Mr. Romney was in “full control of Bain Capital through 2002” and directly responsible for the related outsourcing, bankrupting and layoffs.

“Whatever the consequences of Bain investments were, he was responsible,” she said.

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