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Charges of accepting foreign money also dogged the 1996 re-election bid of Bill Clinton and Mr. Obama’s 2008 campaign.

Several fundraisers - including James Riady, Maria Hsia, Johnny Chung, John Huang and Charles Yah Lin Trie - were convicted of, or pleaded guilty to, charges related to foreign money being funneled to the Clinton-Gore campaign or other Democratic groups during the 1996 election cycle.

In 2008, the Obama campaign did not use simple computer-verification technology to pre-screen Internet donations, including for whether they came from abroad or from U.S. citizens who hadn’t already donated the maximum legal amount. Instead, it attempted to verify the donations after they were deposited, which meets federal-law requirements. Some money was returned, and no criminal charges were filed.

Mr. Gillespie also noted, even apart from legal issues of disclosure, Mr. Obama was elected with no small boost from money that need not legally be disclosed.

“You know, $400 million was spent in 2008 to help elect Barack Obama. We didn’t hear Mr. Axelrod or others complaining about that. Much of that money was undisclosed. And this year, there are organizations on the right who are playing by the same rules,” Mr. Gillespie said.