A push to rename a Senate office building after the late Sen. John McCain John Sidney McCainAnalysis: Biden victory, Democratic sweep would bring biggest boost to economy The Memo: Trump's strengths complicate election picture Mark Kelly: Arizona Senate race winner should be sworn in 'promptly' MORE (R-Ariz.) is facing political headwinds on Capitol Hill from Republican lawmakers.

Sens. Charles Schumer Chuck SchumerPelosi slams Trump executive order on pre-existing conditions: It 'isn't worth the paper it's signed on' 3 reasons why Biden is misreading the politics of court packing Cruz blocks amended resolution honoring Ginsburg over language about her dying wish MORE (D-N.Y.) and Jeff Flake Jeffrey (Jeff) Lane FlakeHow fast population growth made Arizona a swing state Jeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Republican former Michigan governor says he's voting for Biden MORE (R-Ariz.) are working on a resolution that would rename the Russell Senate Office Building, named in honor of former Democratic Sen. Richard Brevard Russell (Ga.), after McCain.

But the proposal is facing skepticism, and in some cases outright opposition, threatening to inject partisan fighting into the chamber's bipartisan mourning.

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“He was a well-respected senator,” Shelby said.

Russell is considered controversial because of his opposition to civil rights legislation.

Shelby noted that if reporters were going to judge Russell on his civil rights stance then they would also need to reevaluate the Founding Fathers.

"If you want to get into that you have to get into George Washington and Thomas Jefferson and all of our — most of our Founding Fathers, maybe with the exception of Hamilton," he said. "It's easy to prejudge what they should have done."

Shelby was one of several Southern GOP senators who pushed back on the idea of taking Russell’s name off the Senate office building.

Georgia Sen. David Perdue — a Republican and close ally of Trump — touted Russell's Senate work, saying on Tuesday that he was a "stalwart" of the military and involved in the Great Society, referring to the domestic program of former President Lyndon B. Johnson.

"This is a guy who was a giant of the Senate,” Perdue said. “So this renaming thing because of one issue, you know, is somewhat troubling. The fact that it's been brought into this John McCain thing I think is inappropriate."

Sen. Bill Cassidy William (Bill) Morgan CassidyCoushatta tribe begins long road to recovery after Hurricane Laura Senators offer disaster tax relief bill Bottom line MORE (R-La.) told reporters that he would prefer to "find another way" to honor McCain.

"What I don't want is to establish a precedent so that something named after John McCain is named after somebody else in the future," Cassidy said.

The GOP pushback is the latest sign that what had emerged as a bipartisan idea in the immediate wake of McCain’s death is already losing momentum in the Senate, where McCain served for more than three decades.

Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellTrump 'no longer angry' at Romney because of Supreme Court stance On The Money: Anxious Democrats push for vote on COVID-19 aid | Pelosi, Mnuchin ready to restart talks | Weekly jobless claims increase | Senate treads close to shutdown deadline The Hill's Campaign Report: Trump faces backlash after not committing to peaceful transition of power MORE (R-Ky.) demurred on Monday when asked if he supports renaming the building after McCain.

He announced on Tuesday that he would create a bipartisan “gang” to make a decision on how the Senate honors McCain.

“The Senate is eager to work on concrete ways to ... provide a lasting tribute to this American hero long after this week's observances are complete," McConnell said.

But he did not include renaming Russell among the possible options, instead using his Senate floor speech to highlight possibly renaming the Armed Services Committee room or hanging a McCain portrait in a room off the Senate floor.

Spokespeople for McConnell didn't respond to a request for comment about whether they were given a heads-up before Schumer said on Saturday, shortly after McCain's death was announced, that he would try to rename the Senate office building.

Flake said he and McConnell were discussing the issue. Schumer directed questions to the Senate GOP leader when he was asked about McConnell's support on Monday.

Schumer told reporters on Tuesday that he didn’t know why some Republicans were reluctant to endorse his idea.

"I think it's the most appropriate way to honor Senator McCain,” he said. “And we're going to work to try and see that that can get done in a bipartisan way."

Flake and Schumer are circulating a "dear colleague" letter asking senators to support the resolution. They have not publicly specified when they would introduce the proposal.

"Renaming the Russell building in his honor ensures that his story will be told with the hope that inspiration becomes motivation and future leaders emerge with his brand of courage and commitment," Flake and Schumer wrote in the letter to their Senate colleagues.

The idea has bipartisan support from Democrats, as well as some Republicans.

Independent Sen. Angus King Angus KingHopes for DC, Puerto Rico statehood rise Government watchdog recommends creation of White House cyber director position Democrats step up hardball tactics as Supreme Court fight heats up MORE (Maine), who caucuses with Democrats, joked on CNN that when he was a Senate staffer people affectionately referred to Russell “as the old SOB."

"And I can’t imagine a more appropriate place to put John McCain’s name," he said on Tuesday.

GOP Sen. Bob Corker Robert (Bob) Phillips CorkerHas Congress captured Russia policy? Tennessee primary battle turns nasty for Republicans Cheney clashes with Trump MORE (Tenn.) — who, like McCain, was critical of Trump on foreign policy — told reporters that he was open to "whatever" to honor McCain.

"Whatever it is appropriate," Corker told reporters, throwing up his arms when asked specifically about renaming Russell. "I'm up for whatever."

GOP Sen. Lindsey Graham Lindsey Olin GrahamSteele Dossier sub-source was subject of FBI counterintelligence probe Hillicon Valley: Subpoenas for Facebook, Google and Twitter on the cards | Wray rebuffs mail-in voting conspiracies | Reps. raise mass surveillance concerns Key Democrat opposes GOP Section 230 subpoena for Facebook, Twitter, Google MORE (S.C.), a close friend of McCain's, joked with reporters that he would rename the Capitol, the Capitol Visitors Center and the Pentagon after McCain if he could, before acknowledging he wanted to get feedback from McCain's family.

"I'd name the Capitol after him, if I could," he told reporters. "But I don't know, I want to talk to Cindy and see what they think."

Several GOP senators pointed on Tuesday to McConnell's decision to form a committee as the best path forward.

Perdue said he supported McConnell’s decision to form a committee and that it was “premature” to discuss renaming Russell, one of three Senate office buildings on Capitol Hill.

"I'm predisposed to say that renaming that building is a serious issue, but I believe right now it is even premature to talk about that,” he said when asked if he would support renaming the building if that is what the panel recommends.

Sen. Johnny Isakson Johnny IsaksonLobbying world Loeffler paints herself as 'more conservative than Attila the Hun' in new campaign ad Georgia GOP Senate candidates cite abortion in pushing Ginsburg replacement MORE (R-Ga.) said McConnell’s panel was a “great idea” but declined to comment on Schumer’s resolution. Sen. John Thune John Randolph ThuneSenate passes resolution reaffirming commitment to peaceful transition of power Susan Collins faces political land mine with Supreme Court fight Abortion stirs GOP tensions in Supreme Court fight MORE (R-S.D), asked if he would support renaming the building, pointed to the process set up by McConnell.

"Well, I'm very much in favor of honoring Senator McCain,” Thune said. “How we do that exactly is something I think is something that we're going to have a conversation about.”

Molly Hooper contributed.

Updated at 5:06 p.m.