At the same time, the White House released a statement from the president in which Trump said that, despite differences on policy and politics, he respects McCain's service to the US and ordered flags be flown at half-staff until his internment.
McCain, a six-term Arizona senator who died at 81 after a long battle with brain cancer, was widely respected for his history as a Vietnam war hero and his commitment to his moral code over party alliance.
And Senator Susan Collins, who many have looked to resist the president's most extreme tendencies, appeared to be uncomfortable when his torch was offered to her.
"I'll do my best", Graham said. How do you accurately be able to give somebody their due justice who served the United States for 60-plus years, I mean dedicated his entire life? McCain's did the same, reminding Americans that for all our differences and sometimes "vociferous" debates, "we have always had so much more in common with each other than in disagreement". Following McCain's death on Saturday, the White House had flown its flags at half staff on Sunday, and then raised them to full staff minutes after midnight on Monday morning.
The White House flag will fly at half-staff for the rest of the week in honor of Sen.
Vice President Mike Pence will speak at a ceremony for McCain on Friday at the U.S. Capitol.
John McCain's brother calls death an 'incredible shock'
On the floor of the Senate, Republican Senator Johnny Isakson of Georgia warned against anyone speaking ill of John McCain. Graham and McCain were two good friends who worked alongside each other in the Senate for more than 15 years.
"Each day, for more than 2,000 days, he endured horrors that few of us could imagine", Hatch said.
The Senator's funeral services later in the week will be high profile, with former presidents George Bush and Barack Obama delivering eulogies before he is buried in a private service at the United State Naval Academy Cemetery in Annapolis, Maryland on Sunday.
But McCain and Trump were never able to bridge the gap between them. Trump's antipathy for McCain bordered on the personal.
Senate Majority Leader Mitch McConnell is hitting the brakes on a push to rename the Senate's oldest office building for the late Sen.
Keep in mind, in John McCain we're talking about someone who was unusually successful in making friends and working with people across the political spectrum.
"To those who are striving as a young person, remember John McCain", he said.
"But good, decent, honorable is John McCain", he said.
The Oklahoma senator called McCain a "hero" and a "patriot" and praised his service.