Senator Graham has come under fierce criticism after the Georgia secretary of state revealed that the politician had hinted he should toss out legal ballots.
Mr Graham told reporters on Tuesday that he had also spoken to election chiefs in Arizona and Nevada out of concern for voting integrity.
But his claim was immediately rejected by the Arizona secretary of state Katie Hobbs.
“This is false. I have not spoken with @lindseygrahamsc,” she wrote on Twitter.
— Secretary Katie Hobbs (@SecretaryHobbs) November 17, 2020
Georgia secretary of state Brad Raffensperger says Mr Graham hinted he should get rid of legal ballots in the state where a recount is underway after Joe Biden won.
Fraudulently altering a federal election vote tally is a federal crime.
Mr Graham has followed in the footsteps of the outgoing president by claiming ballot fraud in the 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.’
Mr Raffensperger says that Mr Graham had cast doubt on the state’s signature matching law and suggested that biased poll workers could have counted ballots with non-matching signatures.
“He asked if the ballots could be matched back to the voters,” Mr Raffensperger said of Mr Graham.
“And then he, I got the sense it implied that then you could throw those out for any, if you look at the counties with the highest frequent error of signatures. So that's the impression that I got.”
Mr Raffensperger has come under attack from Donald Trump on Twitter and faced calls to resign from Georgia’s Republican senators.
Mr Biden has been projected to win Georgia and its 16 electoral college votes by around 14,000 votes.