Senator Graham came under increasing pressure after an official overseen the call, where he asked the Secretary of State about the possibility of throwing a legal ballot.
The vice-president of the Congress Progressive Caucasus has now called on Democratic Representative Pramila Jaipal to stop Mr Graham from interfering.
“Lindsay Graham must resign,” he tweeted.
“It simply came to our notice then that he had asked Georgia’s secretary of state to find a way to legally discard the ballots.
“There must be accountability and justice for this dangerous attack on our democracy.”
A New York Democrat attacked Mr. Graham’s actions.
“Lindsay Graham appears to have invalidated the line in an attempt to snatch Georgia’s Joe Biden’s victory, and this is probably part of a plan to try to steal this election elsewhere,” said Rep. Judge Jeffries.
Civil rights leaders have criticized Mr Graham for his actions and called on Senate committees to launch an immediate investigation.
In Georgia, a countdown is underway to overthrow the outgoing president after defeating the outgoing president by nearly 14,000 votes and winning one of his electoral electoral colleges.
A top aide to Georgia’s Secretary of State Brad Rafensperger confirmed that when Mr Graham asked the state election chief if he had the power to drop some missing ballots.
Mr. Rafensperger said he accepted the advice of his fellow Republicans that he should intervene in the race.
“It simply came to our notice then that he wanted to take to the streets,” Mr Raffensperger told The Daily The Washington Post Earlier this week.
Mr Graham, who donated 500 500,000 to overturn the election result, denied the allegations, calling them “ridiculous”.
He says the call was actually about tightening the voter ID law before the Georgia Senate race in January.
Mr Sterling, who oversees Georgia’s voting system, said in a statement that Mr Graham’s comments on what he thought were acceptable to the public had helped him “maybe go a little further”.
Mr Raffensperger said Mr Graham, one of Donald Trump’s vocal allies, was skeptical of the state’s signature matching law and suggested that biased poll workers could count ballots with unmatched signatures.
Mr Graham claimed he had also called election bosses in Arizona and Nevada, expressing concern for the integrity of elections across the country.
But the claim was quickly shot down by Arizona State Secretary Katie Hobbs, Arizona Secretary of State.
Republican Nevada Secretary Barbara Segavaske also declined to speak with Mr. Graham.
He later clarified that he had spoken with the governor of Arizona but could not recall who he had contacted in Nevada.