Barack Obama, who beat Mr McCain to the White House in 2008, said they shared a "fidelity to something higher".
George W Bush described him as "a patriot of the highest order".
President Donald Trump, whom Mr McCain had strongly criticised, tweeted his "deepest sympathies" to Mr McCain's family but did not comment on his life.
Sources quoted by US media said Mr Trump would not be invited to the funeral and the current administration would probably be represented by Vice President Mike Pence.
Former Presidents Obama and George W Bush are expected to give eulogies.
Mr McCain was diagnosed with an aggressive brain tumour in July last year and had been undergoing treatment.
But on Friday his family announced he had decided to discontinue that treatment.
A statement from his office on Saturday said Mr McCain died in Arizona surrounded by family members.
His family said he would lie in state in Phoenix, Arizona, and in Washington DC before a funeral at the Washington National Cathedral and his burial in Annapolis, Maryland.
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