Washington (CNN) On the same night Sen. John McCain's death was announced, Senate Minority Leader Chuck Schumer said he wanted to change the name of the Russell Senate Office Building after his friend.

But while beloved by his colleagues, Republican senators were mixed in their support for the idea on Monday, as they wandered into the Senate on the first day of session since McCain's passing -- his desk draped in a black cloth and topped with white roses.

Schumer's plan to introduce a resolution left senators scrambling Monday evening as they returned. Georgia Democratic Sen. Richard Brevard Russell, whom the Senate building is currently named for, was a segregationist who was opposed the Civil Rights Act. Russell served from 1933 to 1971 and helped write the Southern Manifesto, which outlined congressional opposition to racial desegregation.

Republican leaders insist they want to honor McCain's more than three-decades long legacy in the Senate in a serious way, but many cautioned it was too soon to decide exactly how to commemorate his service. Underlying the discussion is the residual animosity between McCain and President Donald Trump, who only praised the Vietnam War veteran's service and sacrifice 48 hours after his death was announced.

Too soon?

After he delivered a stirring tribute to McCain on the floor, Majority Leader Mitch McConnell was asked about naming the building after McCain and said only "we'll be talking about the best ways to honor Sen. McCain in the days ahead."

Sen. Roy Blunt, a Missouri Republican, said he's open to Schumer's suggestion.

"I am open to that. I think we ought to have a little discussion about changing the name of a building. But certainly John McCain should be recognized in some way. That's been suggested and it would come through the Rules Committee, I would think," said Blunt, chairman of Rules Committee. "And I think we should actually let at least this week pass before we start talking about how we're going to memorialize Sen. McCain."

John Thune, the No. 3 Senate Republican, said senators were "very interested in honoring Sen. McCain," but "I don't know about the specific idea that has been put out there."

"I am sure we will want to do something. It's something we'll want to talk about," Thune said.

'McCain was a hero'

Others were more open to Schumer's proposal.

Sen. Orrin Hatch, a Utah Republican who is not running for re-election, said he thought renaming the Russell office building would be a good idea.

"I would be glad to do that. I think, you know, Russell was a powerhouse here but most people don't know who he was. And McCain? McCain was a hero whether you liked some of his approach is or not. I liked them and I'd be for that," said Hatch, currently the longest serving Republican senator holding office.

Sen. Tim Scott said that he would also back the idea.

"John McCain has served his country in many facets and a building named after him makes a lot of sense to me," said Scott, the only black Republican senator. "I think the fact of the matter is a prisoner of war for five years who decides to stay there, someone who served in Congress, served in the United States Senate, someone who believed in American exceptionalism. Sounds like a good idea."

It's complicated ...

JUST WATCHED Trump finally speaks about John McCain Replay More Videos ... MUST WATCH Trump finally speaks about John McCain 01:36

The question of whether to rename the Russell Senate office building for McCain is a complicated one for Republicans, putting them at odds with a President who has carried on an adversarial relationship with McCain from the early days of his campaign.

"He's not a war hero. He's a war hero because he was captured. I like people that weren't captured, OK?" Trump said on the campaign trail. "I hate to tell you. He's a war hero because he was captured. OK?"

The relationship between McCain and Trump continued to be tested in the months ahead. In July 2017, McCain voted against a narrow bill that would have advanced Trump's pursuit to repeal the Affordable Care Act, a move that Trump blasted McCain for at rallies even as the Arizona battled cancer.

While Trump's signature wouldn't be required for a Senate resolution, the tug of war Republicans may feel could be a factor.

Russell's defender

Sen. David Perdue, a Georgia Republican who holds Russell's seat, defended the building's current namesake.

"This was an icon in the United States Senate. He was Lyndon Johnson's close adviser. They did the Great Society together. So, people would criticize Richard Russell for maybe being on the wrong side of the integration movement, but my goodness he turned around and got the school lunch program done. He did that himself," Perdue said, adding that Russell was also a stalwart on the Senate Armed Services Committee.

Perdue said he would be opposed to moving forward right now with renaming Russell after McCain and would object on the floor if a member moved to do that. He noted it was hypothetical to assume someone would at this point.

Perdue said he talked to Trump Monday morning about spending bills, judicial nominees and McCain's passing, but he would not comment on if Trump talked with him about renaming the Russell building.

"We talked about John McCain passing. Let's just leave it at that, OK?" Perdue said.