Republican Sen. Chuck Grassley of Iowa, who served with McCain for 31 years, said Monday he was in favor of "naming nearly any building" for McCain, but added, "I'm not sure that I would make a decision on a specific building at this point".
On Monday morning, federal employees received word that they were to immediately lower flags at federal properties around the country to half-staff to honor Sen.
Following Mr McCain's death, admirers have recalled with fondness his insistence on fair and civil discourse - in contrast with the frequent incivility of Mr Trump and his habit of coining unflattering descriptions of political foes. And his broadcast on Radio Hanoi had been played in the prison cell of John McCain.
Many criticized that decision, including the American Legion. The days after someone's passing are not the right time to vent or settle scores. "What I don't want is to establish a precedent so something named after John McCain is named after somebody else in the future".
Graham's tears were the rawest point in a day in which members were still coming to grips with a Senate now permanently without McCain's wily, white-haired presence, something that had been a constant feature of the institution for more than 30 years.
"I do not cry for a flawless man", Graham said.
Graham also pointed out times when McCain's differences with his own party over policies turned into public clashes - all without mentioning the bad blood between McCain and President Trump that persisted even after McCain died. "This week is about John McCain and his legacy [.] You can get into the fight between the president and John McCain, I'm not going to". Before his death, McCain decided Trump should not be invited to his funeral - where former Presidents Barack Obama and George W. Bush are expected to speak.
McCain will lie in state at the Arizona State Capitol and at the U.S. Capitol in Washington, D.C. before his burial on Sunday at the U.S. Naval Academy.
Few returning Republicans have displayed much appetite for taking on McCain's role as a legislator willing to openly challenge the scandal-plagued president.
British Prime Minister Theresa May said McCain "embodied the idea of service over self". He said McCain was considered a war hero "because he was captured" and said he liked "people that weren't captured".
After reciting what he called "dumb jokes" between the two, Graham went further in describing the type of man the Arizona senator was. We believe always in the promise and greatness of America because nothing is inevitable here. "He failed a lot, but he never quit".