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.
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."
Copyright 2016 Nexstar Broadcasting, Inc. All rights reserved. This material may not be published, broadcast, rewritten, or redistributed.