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.) on Tuesday said that he is forming a bipartisan committee tasked with determining how the Senate will honor the late GOP 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 (Ariz.).

"I'd like to put together an official group that can elaborate and bring together ideas from current members, former colleagues and friends. It will be bipartisan as only benefits John's legacy, and come to think of it, [I] would probably call it not a committee but a gang," McConnell said from the Senate floor, referencing McCain's participation in groups including the "Gang of 14" and the "Gang of Eight."

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McConnell later told reporters that he'd be “appointing a group on a bipartisan basis to convene after Labor Day and to think thoroughly through the appropriate way to honor our colleague."

Though McCain's death has sparked bipartisan mourning on Capitol Hill, where the 81-year-old senator was deeply respected by both parties, senators have yet to rally around the best way to remember him.

"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 on Tuesday.

The Senate GOP leader noted that some have suggested naming the room of the Senate Armed Services Committee, which McCain chaired, after the GOP senator.

Others, he said, have suggested putting McCain's portrait in a room directly off the Senate chamber, where only seven senators have their portraits hung.

"It's a further tribute to our colleague that there's no shortage of good ideas," McConnell said.

But Democrats, as well as some Republicans, are lining up behind renaming the Senate Russell Office Building after McCain. Senate Minority Leader 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 GOP Sen. 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 (Ariz.) are drafting a resolution to rename the building.

But GOP senators have been lukewarm to the idea.

McConnell demurred on Monday when asked if he would support it, noting senators would be discussing how to honor McCain.

Sen. John Cornyn John CornynQuinnipiac polls show Trump leading Biden in Texas, deadlocked race in Ohio The Hill's Campaign Report: GOP set to ask SCOTUS to limit mail-in voting Liberal super PAC launches ads targeting vulnerable GOP senators over SCOTUS fight MORE (R-Texas), the No. 2 Senate Republican, said on Monday that there would likely be a “discussion” and suggested the Rules Committee could take up the issue.

- Alexander Bolton contributed