(CNN) -

More than a week after results declared his defeat, the tea party-backed candidate in Mississippi's GOP primary for the U.S. Senate notified Sen. Thad Cochran on Thursday that he plans to challenge the outcome.

State Sen. Chris McDaniel's campaign served papers to Cochran's son, Clayton, with notice of intent to challenge the results, citing allegations of improper crossover voting, according to the Clarion-Ledger of Mississippi.

McDaniel's team had dispatched volunteers across the state to investigate election results in the state's 82 counties. An outside group has already filed a lawsuit in federal court.

McDaniel's campaign has retained a legal team and is urging supporters to donate to the campaign legal fund.

The challenge was filed with the state Republican Party executive committee, as required by law, the Clarion-Ledger reported, and an official court challenge could come as early as next week.

Campaign staffers and 150 volunteers have already combed through voting records in 51 counties and claim to have identified nearly 5,000 "irregularities," which are mostly tied to people who were ineligible to vote in the state's June 24 runoff election, McDaniel campaign spokesman Noel Fritsch said.

The McDaniel campaign announced earlier Thursday that it is offering 15 rewards of $1,000 each for individuals who "provide evidence leading to the arrest and conviction of anyone involved in voter fraud." Donations to the campaign's legal coffers will help fund the rewards.

McDaniel supporters called foul after Cochran's campaign and allies turned to African-Americans and other traditionally Democratic voters to help push the incumbent to victory in the primary runoff.

Cochran had finished about 1,500 votes short of McDaniel in the primary three weeks earlier, but a third candidate kept McDaniel under the 50% threshold needed to win outright.

Voters who cast a ballot in the Democratic primary on June 3 were not allowed to vote in the Republican primary runoff last week.

Predicting ineligible votes, conservative groups hired former Justice Department official Christian Adams to train and oversee election observers to monitor polls and note questionable voter activity.

Adams declined to comment for this story and deferred to the groups he is working with, like FreedomWorks, which he said he may represent in a potential legal challenge.

FreedomWorks Executive Vice President Adam Brandon said the group's activists have been helping the McDaniel campaign review election results and will support the campaign if it moves forward with its intent to challenge -- which could come as soon as early next week.

"We'll ask some of our members to raise the necessary funds to fund such a challenge, and we're standing by and ready to go, period," Brandon said. "It's now up to the folks in Mississippi to figure out what the next steps are."

Challenger's camp: 'irregularities' growing

The number of inadmissible votes the McDaniel camp said it discovered jumped from about 3,300 to nearly 5,000 on Wednesday after the campaign reviewed records in a dozen more counties.

The irregularities fall short of Cochran's nearly 7,000-vote margin of victory, but McDaniel's campaign is confident it will hit that number after reviewing the results from the remaining third of Mississippi counties.

Shortly before a bizarre turn of events Wednesday when McDaniel supporters crashed a Cochran conference call with reporters, Cochran campaign spokesman Austin Barbour called out the McDaniel camp for drawing out a divisive primary battle.

"I just think the time has come now for the McDaniel campaign to put up or shut up," Barbour said at a news conference Wednesday and later that

Fritsch criticized Barbour's remarks in an interview with CNN and said he hopes those are not the views of the Cochran campaign.

"I find it troublesome that Austin Barbour would call for a movement being led by the people, which is in the pursuit of truth and in the pursuit of the maintenance of electoral integrity, to shut up," Fritsch said.

The McDaniel campaign can only file a formal complaint after the state GOP sends certified election results to the Mississippi secretary of state, according to the state party's communications director, Bobby Morgan.

The Mississippi Republican Party's executive committee and representatives from both campaigns met Tuesday, but the committee did not certify election results, Morgan said, adding that just over half of the counties had so far submitted their results.

Morgan said the state party has followed the law throughout the election.

"We're looking forward to the fall campaign. We want to resolve this as soon as possible," he said. "Republicans have a real legitimate chance to retake the Senate and we want to do all we can to make sure the (Mississippi) seat remains in Republican hands."