Senate Republicans are cautioning Sen. Lindsey Graham Lindsey Olin GrahamLincoln Project mocks Lindsey Graham's fundraising lag with Sarah McLachlan-themed video The Hill's Morning Report - Sponsored by Facebook - Republicans lawmakers rebuke Trump on election Trump dumbfounds GOP with latest unforced error MORE (R-S.C.) against the prospect of bringing former Vice President Joe Biden Joe BidenFormer Pence aide: White House staffers discussed Trump refusing to leave office Progressive group buys domain name of Trump's No. 1 Supreme Court pick Bloomberg rolls out M ad buy to boost Biden in Florida MORE before the Senate Judiciary Committee to testify about Ukrainian corruption, saying such a move could backfire.

Graham initially ruled out the possibility of having Biden or his son Hunter testify before the Judiciary panel about their actions related to Ukraine, but he is now open to the idea, which is being pushed by President Trump Donald John TrumpFederal prosecutor speaks out, says Barr 'has brought shame' on Justice Dept. Former Pence aide: White House staffers discussed Trump refusing to leave office Progressive group buys domain name of Trump's No. 1 Supreme Court pick MORE’s personal lawyer Rudy Giuliani Rudy GiulianiThe Hill's Campaign Report: GOP set to ask SCOTUS to limit mail-in voting CIA found Putin 'probably directing' campaign against Biden: report Democrats fear Russia interference could spoil bid to retake Senate MORE.

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Senate Republican Whip John Thune John Randolph ThuneHouse to vote on resolution affirming peaceful transition of power Trump dumbfounds GOP with latest unforced error Senate passes resolution reaffirming commitment to peaceful transition of power MORE (S.D.), the No. 2 Senate GOP leader, on Wednesday said it’s ultimately up to Graham whether to request the Bidens answer questions about whether they played any role in shielding a Ukrainian gas company from a corruption investigation.

But Thune warned that Graham needs to be careful about following Giuliani’s lead on investigating Biden.

“If you’re Lindsey, I’m not sure I’d be taking recommendations from Rudy Giuliani on who I bring” in front of the committee, Thune said.

Giuliani has lost some credibility with GOP lawmakers after two associates who helped him compile information on Ukrainian corruption were arrested at Washington Dulles International Airport last week trying to leave the country.

Sen. John Cornyn John CornynHillicon Valley: Productivity, fatigue, cybersecurity emerge as top concerns amid pandemic | Facebook critics launch alternative oversight board | Google to temporarily bar election ads after polls close Lawmakers introduce legislation to boost cybersecurity of local governments, small businesses On The Trail: Making sense of this week's polling tsunami MORE (R-Texas), a senior member of the Judiciary Committee, said the panel should focus on matters other than the Bidens.

“It wouldn’t be my highest priority. We need limited bandwidth if we need to try to focus on getting things done, not contributing to the sideshow,” Cornyn said.

Sen. Chuck Grassley Charles (Chuck) Ernest GrassleyGOP lawmakers distance themselves from Trump comments on transfer of power The Hill's 12:30 Report: Ginsburg lies in repose Top GOP senators say Hunter Biden's work 'cast a shadow' over Obama Ukraine policy MORE (R-Iowa), the previous chairman of the Judiciary Committee, said he would want to know exactly why Joe Biden needs to appear before the panel, seeing as how the former vice president is a leading candidate to win the Democratic nomination to face Trump next year.

“I’d want to know what he wants to accomplish by bringing him before the committee,” Grassley said of Graham.

On Wednesday, Graham said he hadn’t made a decision on whether to call Joe or Hunter Biden before his committee, something Giuliani says is essential to pushing back against the impeachment effort by House Democrats.

“I don’t know yet,” Graham said. “Right now I want to see what the Intel Committee does.”

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“I’m concerned about the corruption in the Ukraine. I’d like to hear from Rudy,” Graham said, calling Giuliani the “foundational witness.”

Graham added that he might subpoena Giuliani if he declines an invitation to voluntarily appear before the Judiciary Committee.

“I’m going to ask him. If I see a need to subpoena him, I will,” Graham said, adding that Giuliani would testify in an open session of the committee.

The remarks came amid an uncharacteristic public spat between Trump and Graham over the president’s decision to withdraw U.S. troops from northern Syria.

At one point, Trump said Graham should focus on investigating the 2016 election, with a focus on Obama administration officials. The president suggested he look into former President Obama, former CIA Director James Brennan, former FBI Director James Comey James Brien ComeySteele Dossier sub-source was subject of FBI counterintelligence probe Judge will not dismiss McCabe's case against DOJ Democrats fear Russia interference could spoil bid to retake Senate MORE and others, alleging they acted corruptly in the lead-up to his electoral victory.

“That’s what Lindsey ought to focus on,” Trump said. “That’s what the people of South Carolina want him to focus on.”

Whether the fallout of that exchange will extend to possible Biden testimony is yet to be seen.

Giuliani says that Biden and his son, who was paid as much as $50,000 a month to serve on the board of Burisma Holdings in Ukraine, should be called before Congress as part of the GOP’s defense of Trump.

Giuliani argues that Trump was acting in the nation’s interest when he asked Ukrainian President Volodymyr Zelensky in a July 25 phone call to investigate whether Biden played any role in thwarting a corruption investigation in Ukraine, which Giuliani believes was motivated by a desire to protect his son.

“If I were in the House and I had subpoena power, my first witness would be Joe Biden for Trump because I would want to show the corruption that he unleashed in Ukraine that deprived us of the information that could have exonerated the president,” Giuliani told The Hill last week. “My second witness would be Hunter.”

Trump and Giuliani have repeated unsubstantiated allegations that Biden acted corruptly in dealing with Ukraine. Giuliani believes Biden used his power as vice president to get a Ukrainian prosecutor fired to protect his son.

Graham previously dismissed the idea of bringing Joe Biden before his panel, telling reporters that he didn’t “want to turn the Senate into a circus.”

“I have no interest in opening up that front. I don’t want to blow up the Senate,” he said last month.

But he changed his mind over the two-week October recess in reaction to what he viewed as an intensification of partisan tactics by House Democrats and their impeachment inquiry into Trump.

Graham told Fox News’s Bret Baier on Tuesday that he would “look at” calling Hunter Biden to testify if Giuliani first answered questions before the panel.

Several Republican lawmakers have highlighted Hunter Biden’s cushy post as a sign of possible corruption.

“I’ve got some real legitimate questions about what happened with Vice President Biden, Hunter Biden, what was happening over there ... I have a lot of questions that should be answered,” Sen. Ron Johnson Ronald (Ron) Harold JohnsonThe Hill's Morning Report - Sponsored by Facebook - Trump previews SCOTUS nominee as 'totally brilliant' The Hill's 12:30 Report: Ginsburg lies in repose CHC leaders urge Senate to oppose Chad Wolf nomination MORE (R-Wis.), one of Trump’s regular defenders, told reporters last month.

But now there’s also a growing sense among GOP lawmakers that hauling Giuliani and Biden to testify under oath before a Senate committee could turn out to be a bad idea.

For one thing, it could put Giuliani in the uncomfortable position of having to answer questions from Democrats about how much influence he has as Trump’s private lawyer over U.S.-Ukraine policy.

“He’s apparently, unbeknownst to all of us now, carrying out missions for the president of the United States and conducting foreign policy for the U.S. This was one of the many of the surprises of the last two weeks, to find out that Rudy Giuliani and this other guy are carrying out shadow foreign policy for the U.S.,” said Sen. Tim Kaine Timothy (Tim) Michael KaineTrump plans to pick Amy Coney Barrett to replace Ginsburg on court Hillicon Valley: Subpoenas for Facebook, Google and Twitter on the cards | Wray rebuffs mail-in voting conspiracies | Reps. raise mass surveillance concerns Democrats call for declassifying election threats after briefing by Trump officials MORE (D-Va.), referring to Gordon Sondland, a hotelier and Trump donor who now serves as the U.S. ambassador to the European Union and who worked with Giuliani on investigating Ukrainian corruption.

Bringing Biden before the Judiciary Committee might also play into Democratic charges that Trump and his GOP allies have tried to use the power of the federal government to hamstring a political rival.

And it could also further inflame partisan tensions and make it tougher to get regular business done, as Graham said he feared when speaking to reporters last month.

Sen. Richard Blumenthal (D-Conn.) on Wednesday warned that Graham needs to have a very strong reason to call on the Bidens to testify.

“There should be a very strong reluctance to bring a presidential candidate who has no official position right now before the committee without a really strong basis, and I see no basis right now,” he said.

Other Republican members of the Judiciary panel on Wednesday said they would leave things up to Graham.

“I defer to the chairman there on what jurisdiction we have or don’t have,” said Sen. Josh Hawley Joshua (Josh) David HawleyHillicon Valley: Subpoenas for Facebook, Google and Twitter on the cards | Wray rebuffs mail-in voting conspiracies | Reps. raise mass surveillance concerns Trump faces tricky choice on Supreme Court pick FBI director warns that Chinese hackers are still targeting US COVID-19 research MORE (R-Mo.).

Sen. Ted Cruz Rafael (Ted) Edward CruzCrenshaw looms large as Democrats look to flip Texas House seat SCOTUS confirmation in the last month of a close election? Ugly The Hill's Morning Report - Sponsored by Facebook - Trump previews SCOTUS nominee as 'totally brilliant' MORE (R-Texas) said, “That’s a decision for the chairman to make.”

Sen. Joni Ernst Joni Kay ErnstOn The Trail: Making sense of this week's polling tsunami Tillis appears to reinforce question about COVID-19 death toll The power of incumbency: How Trump is using the Oval Office to win reelection MORE (R-Iowa), who is up for reelection next year, said it’s something “that should be up to Graham and Giuliani.”

“I think we’re going to have a lot of time to hear evidence in the upcoming months,” she added.

Jordain Carney contributed.