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(CNN) -- The tea party-backed challenger who narrowly lost a Republican primary runoff in Mississippi to longtime Sen. Thad Cochran says he plans to file a legal challenge contesting those results "any day now."

State Sen. Chris McDaniel said Friday on CNN's "New Day" that "the integrity of the process matters. We believe on that night of June 24 there were thousands of irregularities and we've already found thousands of irregularities in the process."

McDaniel also defended his campaign's offer - announced Thursday - of rewards of $1,000 each for individuals providing "evidence leading to the arrest and conviction of anyone involved in voter fraud."

And McDaniel said "we condemn any racist comments what so ever," when asked by CNN anchor Kate Bolduan about a bizarre Cochran campaign conference call on Wednesday that included borderline racist comments by an unidentified caller who appeared to be a supporter of the challenger's campaign.

Led primary, lost runoff

McDaniel narrowly edged Cochran in a June 3 primary, but with neither man cracking the 50% threshold needed to win, the contest moved to the runoff three weeks later, which Cochran won by fewer than 7,000 votes.

His victory was aided by votes from African-American Democrats, who were actively courted during the runoff by Cochran's campaign and allied groups.

According to Mississippi law, voters are not required to register with a political party, and anyone who doesn't vote in a primary election can cast a ballot in either party's runoff.

Since the runoff, McDaniel has refused to concede and has repeatedly vowed to use every legal maneuver available to fight the results.

McDaniel alleges that some Democrats who voted in the runoff had previously cast a ballot in their party's Senate primary.

"Right now, we have found we have found more than 5,000 irregularities. There are more than 19,000 absentee ballots we still haven't seen yet," McDaniel told CNN.

Disputed claims

The Cochran campaign disputes McDaniel's claims, and numbers.

"The time has come now for the McDaniel campaign to put up or shut up. If they have hard evidence, bring it forward. But quit talking about exaggerated numbers that they know are not true," Cochran campaign spokesman and adviser Austin Barbour said Wednesday.

After last week's runoff, McDaniel's campaign dispatched volunteers across Mississippi to investigate the results in the state's 82 counties.

FreedomWorks, one of the anti-establishment groups that's been supporting McDaniel, dispatched activists to assist the campaign.

Separately, a conservative outside group filed a lawsuit in federal court asking for full access to the voting records in the primary and runoff elections.

Won't drag out

McDaniel said on "New Day" that the process "won't drag out too much longer. We have our people in the field. They'll be working all next week to find the additional irregularities that we believe are out there."

McDaniel added that "after that, we'll have our evidence together and move forward."

And he defended his campaign's use of reward money, "if fraud was committed."

He added, "If vote buying took place as alleged, we have an obligation to find it and root it out once and for all. There's nothing wrong with that process. It needs to happen for the integrity of the election process."

McDaniel pushed back allegations that he's contesting the runoff results to further his political career and pay off campaign debts.

"We don't have any primary debt, not one dime," McDaniel told Bolduan, adding that what he's now doing is "bigger than a campaign, it's bigger than a candidate, it's bigger than me."

'Join us in the process'

And he urged the Cochran campaign to "join us in this process. Why not go out there hand and hand and lets find the corruption if it exists."

But that offer didn't fly with the Cochran camp.

"It's obvious that Chris McDaniel doesn't care about the integrity of the process or he wouldn't be offering thousand-dollar bounties in e-mail fundraising blasts and flat out stating that the election was stolen and the election was a sham without any proof," Cochran spokesman Jordan Russell told CNN.

"So I think he needs to spare me the 'why won't they join us in protecting the integrity of the process.' Chris McDaniel doesn't care anything about the integrity of the process. Chris McDaniel cares about staying in the spotlight."

McDaniel condemns race comments

On Wednesday's Cochran campaign conference call, a person who appeared to be a McDaniel supporter asked "since black people harvested cotton, why is it OK to harvest their votes? Why is it OK to harvest the votes of black people?"

Asked about those comments, McDaniel said "we condemn any racist comments what so ever. But bear in mind, we have no idea who that person is."

He added, "You understand there are people out there we have no control over."

If his legal challenge to the runoff results is turned down by the courts, McDaniel told CNN that "certainly we'll respect the court's decisions. As to endorsing or going forward with Thad Cochran or whoever else, I'm going to have to make that call after all this is said and done."

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CNN Political Editor Paul Steinhauser and Jeremy Diamond contributed to this report.