The nation's largest veterans' service organization urged Trump in a statement Monday to follow longtime protocol following the deaths of prominent government officials.
After facing criticism of his response, the president on Monday somewhat changed his tone on the late senator.
"America will miss her dedicated son, her stalwart champion, her elder statesman", McConnell said as McCain's desk on the Senate floor sat draped in black cloth and adorned with vase of white flowers.
Trump also delayed issuing the customary proclamation for flags to remain at half-staff for longer than the two-day minimum.
Flags elsewhere in the capital had remained at half-staff Monday, and it was widely believed that Trump was demonstrating his continued resentment toward McCain, who provided the pivotal vote preventing Republicans from repealing the Affordable Care Act.
Citing unnamed current and former White House aides, the Post said Sanders, Chief of Staff John Kelly and other senior staff had advocated for a statement that called McCain a "hero".
Mr Trump regularly hit back at his rival, and controversially said Mr McCain was "not a war hero" in 2015.
In his own statement, Trump, who was not invited to attend McCain's funeral, said he had asked Vice President Mike Pence "to offer an address at the ceremony honoring" McCain at the U.S. Capitol on Friday. But it flew at full staff on Monday.More news: Turkey FM: Military solution in Syria's idlib 'will cause catastrophe'
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McCain's office was in the beaux-arts style building, on Constitution Avenue, and it's where the committee he proudly chaired, the Senate Armed Services Committee, meets.
Senate Minority Leader Chuck Schumer, D-N.Y., joked that he and McCain did not get along when Schumer first entered the Senate, but that the two formed a lasting friendship working to pass immigration reform.
McCain was a frequent critic of President Donald Trump on everything from his leadership style to his Russian Federation policy. "John McCain was one of them", the New York Democrat said in a statement Saturday night, just after McCain died, CBS News reports. Trump tweeted condolences to McCain's family but made no reference to the Arizona senator.
The flags at the White House, which were lowered over the weekend to mark the death of Sen.
McCain died on Saturday at 81 after a 13-month struggle with brain cancer.
Should one of the Senate's three office buildings, in the shadow of the U.S. Capitol itself, continue to bear Russell's name?
After Democratic Sen. Ted Kennedy of MA died in 2009, President Barack Obama ordered flags at the White House be flown at half-staff for five days.
By day's end, Trump had done just that, with no immediate explanation from the White House.
U.S. Flag Code states that flags be lowered "on the day of death and the following day for a Member of Congress".