A top aide for Georgia secretary of state Brad Raffensperger was on the line when Mr Graham asked the state’s elections chief if he had the power to throw out some absentee ballots.
Mr Raffensperger says he took that as a suggestion by his fellow Republican that he should interfere in the tight race that saw Joe Biden beat Donald Trump by around 14,000 votes.
“It sure looked like he was wanting to go down that road,” Mr Raffensperger told the Washington Post.
Mr Graham, who has donated $500,000 to overturning the election result, has denied the allegation and branded it “ridiculous”,
He now says the call was in fact about strengthening voter ID laws ahead of the Georgia Senate races in January.
But Gabriel Sterling, who oversees voting systems in Georgia, told reporters he had listened in on Mr Graham’s call to his boss.
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“What I heard was basically discussions about absentee ballots, and if a – potentially – if there was a percentage of signatures that weren’t really, truly matching, is there some point when you get to where you can say, if somebody went to the courtroom, could say, ‘Let’s throw out all these ballots because we have no way of knowing because the ballots are separate,’” said Mr Sterling.
“And that was partially what was going on. I could see how Senator Graham viewed it one way and Secretary Raffensperger viewed it one way.
“But, you know, our job in this state has to follow the law and follow the process as we continue to do.
“There’s no physical ability for this office to do anything along those lines. If somebody wanted to go that route, they could go the court route.”
Mr Sterling added that Mr Graham’s comments “might have gone a little to the edge of” what people would find acceptable, according to CNN.
“The president is going to continue to fight, his supporters continue to fight,” added Mr Sterling.
“Our job is to continue to follow the law, and we were answering process questions… that’s what we were doing on the call.”
Mr Raffensperger says that Mr Graham, one of Donald Trump’s most vocal allies, had cast doubt on the state’s signature matching law and suggested that biased poll workers could have counted ballots with non-matching signatures.
Mr Graham has been accused of meddling in the state’s election but has rejected that claim.
He told reporters that he had also called election bosses in Arizona and Nevada out of concern for election integrity across the country.
But that claim was quickly shot down by Arizona secretary of state Arizona secretary of state Katie Hobbs.
“This is false. I have not spoken with @lindseygrahamsc,” she wrote on Twitter.
Nevada secretary of state, Barbara Cegavske, a Republican, also denied having spoken to Mr Graham.
“I have not spoken with Senator Lindsey Graham or any other members of Congress,” she said.
Mr Graham clarified that he had spoken to the Arizona governor but could not remember who he had contacted in Nevada.
Earlier Mr Raffensperger announced that a forensic examination of voting machines in Georgia showed “no sign of foul play” and that they had not been in any way manipulated.
And he is now overseeing a full hand recount of nearly 5 million votes in the presidential election.
Observers say there is no evidence of voter fraud and the Department of Homeland Security called it the most “secure election in American history”.
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