(CNN) -

The campaign of the tea party-backed challenger who narrowly lost a Republican primary runoff in Mississippi to longtime Sen. Thad Cochran said it's building the evidence needed to legally contest those results.

"The evidence we have found so far, which will in part serve as the basis for a challenge, is promising," Noel Fritsch, spokesman for state Sen. Chris McDaniel, told CNN Monday. Fritsch spoke to CNN as attorneys for the campaign told reporters at a news conference at the Hinds County courthouse in Jackson, Mississippi that they will go through ballot boxes, including absentees, before taking legal action.

Earlier Monday Fritsch told CNN Chief Congressional Correspondent Dana Bash that they've found thousands of examples of voter irregularities, as well as invalid crossover votes from Democrats. The latest comments from the McDaniel campaign came just a couple of hours before the deadline for the state GOP to hand over to the Mississippi secretary of state's office the certified results from the runoff contest.

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.

The Mississippi secretary of state's office announced Monday it accepted the certified election results from both state parties. The official results from the Republican Party showed Cochran with more than a 7,600 vote lead over McDaniel.

His victory was apparently 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.

After the 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.

While Fritsch said there were thousands of voter irregularities, he could not provide an exact number. But he added that the irregularities they've they have found so far do not include any from absentee ballots -- just votes that occurred in person in the runoff. After the runoff, McDaniel's campaign dispatched volunteers across Mississippi to investigate the results in the state's 82 counties.

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."

"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 added.

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."

The Cochran campaign has disputed McDaniel's claims, and numbers. The Cochran campaign said Monday afternoon that it has representatives at all 82 courthouses to monitor the review of the ballot boxes, and that its pleased so far with the results.

"The county-by-county results reported thus far are revealing an extremely low number of crossover votes from the June 24th election. As the process moves forward, the conversation is shifting from wild, baseless accusations to hard facts," Cochran spokesman Jordan Russell told CNN. "As we have said from the beginning, the run-off results are clear: the majority of Mississippians voted for Senator Thad Cochran."