More than a week after the June 24 election, McDaniel has still not conceded. He said Thursday he would pay $1,000 to anyone who provides evidence that Cochran's campaign cheated.
Cochran, 76, is seeking a seventh term in the Senate. In Mississippi, which has become one of the most Republican states, winning the party's nomination is likely to mean winning the election.
Under Mississippi law, anyone can vote in any primary, regardless of registration. But those who voted in the Democratic primary were barred by law from voting in the runoff, which was necessary because McDaniel, who ran slightly ahead of Cochran in the primary, failed to win at least 50 percent of the vote.
In the runoff, Cochran defeated McDaniel by 7,000 votes. His margin may have come from black Democrats who decided the conservative Cochran was the lesser of two evils.
McDaniel charges that many of those voters cast ballots in the Democratic primary.
"Sen. Thad Cochran's campaign allegedly paid $15 per vote to steal the June 24 Republican primary election," McDaniel's campaign website says. "To fight the alleged corruption, we need your help. Just $15 will help us begin to eradicate corruption in Mississippi."
The campaign has been unusually nasty. Several McDaniel supporters have been charged with conspiring to photograph Cochran's bedridden wife in the nursing home where she is a resident. One of the suspects, a Mississippi Tea Party official, recently died in an apparent suicide.
Notable deaths of 2014 [PHOTOS]
Pregnant woman chases thief, then gives birth