The former world heavyweight boxing champion, one of the world's best-known sportsmen, died at a hospital in the US city of Phoenix in Arizona state after being admitted on Thursday.
He was suffering from a respiratory illness, a condition that was complicated by Parkinson's disease.
The funeral will take place in Ali's hometown of Louisville, Kentucky, his family said in a statement.
Ali shot to fame by winning light-heavyweight gold at the 1960 Rome Olympics.
Nicknamed "The Greatest", the American beat Sonny Liston in 1964 to win his first world title and became the first boxer to capture a world heavyweight title on three separate occasions.
He eventually retired in 1981, having won 56 of his 61 fights.
Ali had been suffering from respiratory problems
Crowned "Sportsman of the Century" by Sports Illustrated and "Sports Personality of the Century" by the BBC, Ali was noted for his pre- and post-fight talk and bold fight predictions just as much as his boxing skills inside the ring.
But he was also a civil rights campaigner and poet who transcended the bounds of sport, race and nationality.
Asked how he would like to be remembered, he once said: "As a man who never sold out his people. But if that's too much, then just a good boxer. I won't even mind if you don't mention how pretty I was."
Muhammad Ali was crowned Sportman of the Century in 1999
Ali fought Frazier for a third and final time in the Philippines on 1 October 1975, coming out on top in the "Thrilla in Manila" when Frazier failed to emerge for the 15th and final round.
Six defences of his title followed before Ali lost on points to Leon Spinks in February 1978, although he regained the world title by the end of the year, avenging his defeat at the hands of the 1976 Olympic light-heavyweight champion.
Ali's career ended with one-sided defeats by Larry Holmes in 1980 and Trevor Berbick in 1981, many thinking he should have retired long before.
He fought a total of 61 times as a professional, losing five times and winning 37 bouts by knockout.
Soon after retiring, rumours began to circulate about the state of Ali's health. His speech had become slurred, he shuffled and he was often drowsy.
Parkinson's Syndrome was eventually diagnosed but Ali continued to make public appearances, receiving warm welcomes wherever he travelled.
He lit the Olympic cauldron at the 1996 Games in Atlanta and carried the Olympic flag at the opening ceremony for the 2012 Games in London.
