White House aides had reportedly prepared the statement, which would have lauded Mr McCain for his time in the military and politics.
But President Trump is reported to have stepped in and told his staff that he wanted to post a tweet instead.
His tweet, uploaded on Sunday, did not mention any of Mr McCain’s achievements.
Instead it said: “My deepest sympathies and respect go out to the family of Senator John McCain.
“Our hearts and prayers are with you.”
According to The Washington Post, press secretary Sarah Huckabee Sanders, chief of staff John Kelly and other White House aides wanted to release an official White House statement.
Mark Corallo, a former spokesman for Trump’s legal team, called Trump’s decision “atrocious”.
Mr Corallo told The Washington Post: “At a time like this, you would expect more of an American president when you’re talking about the passing of a true American hero.”
Former Senator McCain was known for his work in the US military during the Vietnam war, where he was taken as a prisoner of war in 1973 and sustained life-changing injuries.
As a result of his injuries, Mr McCain suffered lifelong disabilities in his limbs.
While generally adhering to conservative principles, former senator Mr McCain also had a media reputation as a ‘maverick’ for his willingness to disagree with his party on certain issues.
He previously served two terms in the United States House of Representatives and was the Republican nominee for President of the United States in the 2008 election, which he lost to Barack Obama.
Mr Trump and Mr McCain exchanged fiery words in the past, perhaps most viciously when Trump said McCain was a “war hero only because he got caught,” and that he preferred “people who didn’t get captured”.
Mr McCain died on Saturday after his family announced he would not continue with treatment against aggressive brain cancer.
Arizona Governor Doug Ducey, who will be tasked with finding a replacement for Mr McCain, said of Mr McCain: “We’re most proud to call him a fellow Arizonan.
“Like so many of us, he was not born here, but his spirit, service and fierce independence shaped the state with which he became synonymous.”