Chelsea Clinton was seated next to Thurmond at an inauguration lunch.
McCain is taking fire from both sides over the GOP nominee.
McConnell said it could be taken up under Obama's successor.
"It's time for a special prosecutor to investigate this matter."
A new web ad will be released on Thursday.
Cotton called on the foundation to stop accepting foreign and corporate donations.
But the New Hampshire senator said Trump "clearly says what he thinks."
Sen. Roy Blunt wants financial records of the military sales fund.
The Democratic leader launched a petition calling on Trump to take a U.S. citizenship exam.
Sen. Mark Kirk said Obama is "acting like the drug dealer in chief."