In an interview with Radio Iowa, the presumptive Republican nominee said his finances were legitimate and managed by a blind trust, but didn't tackle head on repeated calls from the Obama campaign to release more tax returns.
"I don't manage them. I don't even know where they are," Romney said off his offshore accounts.
"That trustee follows all US laws. All the taxes are paid, as appropriate," Romney added. "All of them have been reported to the government. There's nothing hidden there."
Romney, who earned a vast fortune in the business world as founder and head of the private investment firm Bain Capital before becoming governor of Massachusetts, has released his 2010 returns and an estimate for 2011.
He told Radio Iowa that Obama was trying to shift the campaign discussion away from his presidential record and onto subjects that show his challenger in a poor light.
"I understand the president's going to try to do anything he can to divert attention from the fact that his jobs record is weak and he has no plan to make things better," Romney said.
Romney's wealth, estimated to be around $250 million, has repeatedly surfaced as an issue during the campaign as Obama tries to paint him as out of touch with ordinary Americans before the November 6 vote.
Much of his fortune lies well hidden in a network of opaque offshore investments including some $30 million in the Cayman Islands, as well as in a corporation established in the 1990s in Bermuda, Vanity Fair magazine reported last week.
Democrats have launched a coordinated assault on Romney's use of offshore tax havens, and Obama touched on the issue himself on Monday evening in an interview with the New Hampshire television station WMUR.
"What's important is if you are running for president is that the American people know who you are and what you've done and that you're an open book," Obama said.
"And that's been true of every presidential candidate dating all the way back to Mitt Romney's father," he added, referring to George Romney, who lost the 1968 Republican nomination battle to Richard Nixon.
Obama's deputy, Vice President Joe Biden, took a more strident tone in prepared remarks for a Hispanic event on Tuesday evening in Las Vegas.
"When his father, George Romney, was a candidate for president in 1968, he released 12 years of tax returns because, as he said, 'One year could be a fluke, perhaps done for show.'
"His son has released one year of his tax returns. Making a lie of the old adage: Like father, like son."
On Air Force One, as Obama traveled to a campaign event in Cedar Rapids, Iowa, his campaign spokeswoman Jen Psaki refused to let up on the issue.
Psaki said the average Iowa family who banks at the corner savings-and-loan would be hard pressed to understand why Romney had millions stashed away in Switzerland and in the Cayman Islands.
"He was running a hedge fund -- you may or may not know this -- for three years with major investments in the Cayman Islands. He has an over $100 million IRA (retirement fund) that's based there," she said.
"There are a lot of questions that people want answers to, and the ball is really in Mitt Romney's court and his team's court to release the tax returns so we can get those questions answered."