Graham told Fox News host Sean Hannity that there was already enough support from Judiciary Republicans to advance Trump's pick to a full vote in the Senate.
"We've got the votes to confirm Justice Ginsburg's replacement before the election," he said. "We're going to move forward in the committee. We're going to report the nomination out of the committee to the floor of the United States Senate, so we can vote before the election. Now, that's the constitutional process."
Graham added that the pick "will be supported by every Republican in the Judiciary Committee," including the vulnerable Sen. Joni Ernst, R-Iowa.
"We've got the votes to confirm the judge — the justice on the floor of the Senate before the election," he said, "and that's what's coming."
President Donald Trump, who earlier revealed plans to announce his nominee by the weekend, has vowed to pick a woman. His shortlist reportedly includes 7th Circuit Appeals Court Judge Amy Coney Barrett, 11th Circuit Appeals Court Judge Barbara Logoa and 4th Circuit Appeals Court Judge Allison Jones Rushing.
Graham's comments came after protesters stormed his Washington residence on Monday in protest of his push to replace the late Justice Ruth Bader Ginsburg. Graham was one of the leading Republicans who rejected considering former President Barack Obama's nominee Merrick Garland ahead of the 2016 election.
"I want you to use my words against me," Graham said at the time. "If there's a Republican president in 2016 and a vacancy occurs in the last year of the first term, you can say, 'Lindsey Graham said let's let the next president, whoever it might be, make that nomination.'"
Graham doubled down on his comment in 2018, vowing that "if an opening comes in the last year of President Trump's term and the primary process has started, we'll wait to the next election."
"It's pretty obvious that if they want an outcome, they'll just destroy anybody's life to keep the seats open," Graham said of Democrats. "They said they tried to destroy Brett Kavanaugh, so they could fill the seat. They were dumb enough to say that. I've seen this movie before. It's not going to work. It didn't work with Kavanaugh."
Graham on Monday wrote in a letter to Democrats on the committee that "I am certain if the shoe were on the other foot, you would do the same."
Graham's push comes as he faces an unexpectedly close re-election race against Democrat Jaime Harrison, who called out the senator for his reversal.
"My grandpa always said that a man is only as good as his word," he tweeted. "Senator Graham, you have proven your word is worthless."
With Trump pushing for a confirmation vote before the election, Graham may be forced to remain in Washington in the critical weeks leading up to his election as a flood of outside money pours into ads calling out his walkback.
Graham's turnaround on the Supreme Court was similar to his reversal on the issue of impeachment. Democrats during the trial played a video of Graham, a former Bill Clinton impeachment manager who supported Trump during his trial, arguing that a president did not have to commit a crime in order to be removed from office.
"It's just when you're using your office in a way that hurts people, you've committed a high crime," he said during Clinton's impeachment.
Ads in South Carolina have also focused on Graham's remarks about Trump when the two faced off in the 2016 Republican primary. At the time, the senator said Trump was a "race-baiting, xenophobic, religious bigot" who should be told to "go to hell."
Graham won his last re-election by 15 points in the deep red state, but Harrison has been running about even in recent polls.
Republicans hold a 53-47 majority in the Senate. Though confirming a Trump nominee will likely fall across party lines, Sens. Lisa Murkowski, R-Alaska, and Susan Collins, R-Maine, have said they would oppose a confirmation before the election.
Graham's change of heart has led to a flood of donations to Harrison, who reported raising $1 million on two consecutive days following Ginsburg's death.