The three-time world heavyweight champion, who had battled Parkinson’s disease for 32 years, was admitted to hospital with a respiratory condition earlier in the week.
His family’s spokesman Bob Gunnell confirmed Ali’s death in Phoenix, Arizona, on Friday evening local time.
The funeral will take place in Ali’s home town of Louisville, Kentucky.
The day Muhammad Ali came to South Shields
A statement from the spokesman said the Ali family “would like to thank everyone for their thoughts, prayers and support” and asked for privacy.
Ali, hailed as “The Greatest”, is survived by his fourth wife Lonnie - whom he married in 1986 - and multiple children, many of whom were reported to have flown to their father’s bedside on Thursday and Friday.
He had been admitted to hospital most recently in early 2015 when he was treated for a severe urinary tract infection initially diagnosed as pneumonia.
At his last public appearances, he looked increasingly frail, including on April 9 when he wore sunglasses and was hunched over at the annual Celebrity Fight Night dinner in Phoenix, which raises funds for Parkinson’s treatment.
Ali has suffered from Parkinson’s for three decades and trembled badly while lighting the Olympic torch in 1996 in Atlanta.
Doctors say the Parkinson’s was probably caused by the thousands of punches Ali took during a career in which he travelled the world for big fights.
Tributes quickly flooded in for Ali, born Cassius Clay, as news of his death broke.
George Foreman, Ali’s friend and rival from the famous “Rumble in the Jungle” fight, said: “We were like one guy - part of me is gone.”
He said he wanted Ali to be remembered as a “brave” humanitarian and not just a boxer.
He said: “Muhammad Ali was one of the greatest human beings I have ever met. No doubt he was one of the best people to have lived in this day and age.
“To put him as a boxer is an injustice.”
He also spoke of Ali’s love for the UK and the way he was treated in the country.
“He loved London. If he had been born and raised in London he never would have changed his name,” he said.
Former heavyweight champion Mike Tyson tweeted: “God came for his champion. So long great one. @MuhammadAli #TheGreatest #RIP.”
Another former heavyweight boxing champion, Evander Holyfield told MSNBC: “I’m glad to have known Ali because when I was a kid, at eight years old, I was told I would be like Ali.
“To take it upon yourself and say, ‘I’m the greatest’, you put yourself in a position for people to take pot shots at you. This is what Ali did. It’s amazing him becoming three-time heavyweight champion of the world. At that time people thought, ‘Who could beat three?
“You have to be stronger to get up from a loss to go on and that’s what Ali proved to be.”
His biographer Thomas Hauser told the BBC that Ali was “beautiful inside and beautiful outside”. He tweeted that Ali’s death was “the passing of a legend”.
Retired Irish boxer Barry McGuigan spoke to the BBC about Ali’s actions inside and outside of the ring, including his role in the civil rights movement and his refusal to fight in the Vietnam war.
He said he was “a remarkable human being”.
“He was hugely inspirational for me and many fighters all over the world,” he said.
“Everybody wanted to box because of him.”
He said: “He was just so amazing in every way.
“More than anything else it was how humble and how brilliantly charismatic he was.
“He was a beautiful looking man, a beautiful looking individual and he had so much compassion.”
He added: “He was the greatest sportsman there has ever been and we were very lucky that he chose boxing.