On Saturday, Mr Graham echoed Georgia Governor Brian Kemp's desire for the state to conduct an audit of mail-in ballot signatures.
Mr Kemp said he would accept the results, but only because he was required to by law.
Donald Trump criticised Mr Kemp for not cracking down on Mr Raffensperger, and claimed without providing any evidence that a signature audit would prove that illegal voting had taken place.
"The Governor of Georgia and Secretary of State, refuse to let us look at signatures which would expose hundreds of thousands of illegal ballots, and give the Republican Party and me, David Perdue, and perhaps Kelly Loeffler a BIG VICTORY," Mr Trump tweeted on Friday.
Mr Graham has been one of Mr Trump's most vocal allies in his campaign to overturn the election by invalidating votes — and has been accused of trying to influence local election officials to help the president achieve that aim.
Mr Raffensperger told The Washington Post earlier in the month that Mr Graham asked him if he could throw out all of the mail in ballots from counties with high rates of mismatched signatures. Later, Mr Raffensperger confirmed that he believed Mr Graham wanted him to throw out legally cast ballots.
Mr Graham denied he asked anything of the sort.
“We never talked about him throwing out ballots,” Graham said. “We talked about how would you challenge a signature in Georgia,” he said during a Fox and Friends interview.
If Mr Graham did ask Mr Raffensperger to throw out the ballots, he would have committed a crime under Georgia election laws. However, Mr Graham is unlikely to face prosecution as he is a sitting senator.
In a tweet on Friday, Mr Graham claimed the rate of signature rejections in the 2020 race was significantly less than in previous elections.
"The rejection rate for signatures in the 2020 race was significantly less than in past elections. A signature audit is necessary given how the Consent Decree affected mail in voting," he wrote.
The allegation that the signature rejection rate is higher is misleading. Mr Trump and Mr Graham's concerns with signature matching are based on a misunderstanding of the data currently available from the state of Georgia.
The state has not released data on the total number of mail-in ballots that have been rejected. It has released the number of mail-in ballots rejected due to missing or mismatched signatures, but those numbers are consistent with the rate of rejections in the 2018 midterms and is lower than the number of rejections in 2016.
While more mail-in ballots were rejected this year than in 2018, more mail-in ballots were also cast in 2020 than in 2018. When taking the volume of mail-in ballots into consideration, the rate is nearly identical to 2018.
Gabriel Sterling, Georgia's voting system implementation manager and a Republican, said the 2020 rejection rate has been "pretty consistent" and said the comparisons Mr Trump and Mr Graham are making are "apples and oranges."