Several Republican senators congratulated Vice President-elect Kamala Harris on her electoral victory, even as many of them publicly rally behind President Trump's efforts to fight the election results.

Sens. Tim Scott, R-S.C., James Lankford, R-Okla., Mike Rounds, R-S.D., and Ben Sasse, R-Neb., all approached Harris to congratulate her Tuesday on the Senate floor as the chamber voted on Federal Reserve board nominee Judy Shelton.

It marked the first time that Harris, a California senator, has appeared on Capitol Hill since becoming vice president-elect. 

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Trump has refused to concede since Fox News and other media organizations projected that President-elect Joe Biden had defeated him in the presidential election. Instead, his campaign is forging ahead with a series of legal challenges against the voting results.

The president's GOP allies have so far stood by him.

Last week, Senate Majority Leader Mitch McConnell, the top Republican in Congress, said that Trump is "within his rights" to challenge the outcome, and criticized Democrats for expecting the president to concede.

“President Trump is 100% within his rights to look into allegations of irregularities and weigh his legal options,” McConnell said. “Let’s not have any lectures about how the president should immediately, cheerfully accept preliminary election results from the same characters who just spent four years refusing to accept the validity of the last election.”

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Just a handful of Republicans, including Sasse, as well as Utah Sen. Mitt Romney and Maine Sen. Susan Collins, have acknowledged Biden as the likely winner of the election.

Sen. Lindsey Graham, R-S.C., -- one of the most vocal defenders of the president -- also gave a fist bump to Harris, though he said later that he was "just saying hello."

"I haven’t seen her in a while. If it works out and they make it, I told her I wish her well and try to work where we can. We will know here in a month or so or less," Graham told CNN.

Graham came under scrutiny earlier in the week after the Georgia Secretary of State Brian Raffensperger, a Republican, accused the senator of personally pressuring him to throw out some legal mail-in ballots during a mandatory recount. 

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Graham has denied the accusations. 

“I know what I talked to the secretary of state about, and it wasn’t for him to throw out ballots, that was beyond bizarre and ridiculous,” Graham said in an interview with Fox News' Sean Hannity on Tuesday. “I was talking about a system and how it works and to make sure we have confidence in it.”