Palmer died at the UPMC Shadyside Hospital in Pittsburgh, Pennsylvania, where he was undergoing heart tests, the Pittsburgh Post-Gazette reported.
During a long career he won more than 90 tournaments worldwide, including seven majors.
The US Golf Association called him “golf’s greatest ambassador”.
As tributes flooded in from across the world of golf, Tiger Woods tweeted: “Thanks Arnold for your friendship, counsel and a lot of laughs. Your philanthropy and humility are part of your legend.”
Palmer attracted thousands of diehard fans known as “Arnie’s army” and helped to promote the game into the television age.
Reaction to golfing great’s death
‘The King’ of golf who changed the game
Obituary: Arnold Palmer
The golfer was loved as an everyman superstar, and even had a drink named after him – the Arnold Palmer cocktail, made from one part iced tea and one part lemonade.
He also gave his name to a professional tournament – The Arnold Palmer Invitational, held each March at his private golf resort in Bay Hill, Florida.Arnold Palmer was the most important golfer to play the game, says the BBC’s golf correspondent Iain Carter.
“His force of charisma put the game on the map and it never dimmed”, he added.
Fellow golfing great and a close friend of Palmer, Jack Nicklaus, said he would “miss him greatly”.
“We just lost one of the incredible people in the game of golf and in all of sports,” he tweeted.
“Arnold transcended the game of golf. He was more than a golfer or even great golfer. He was an icon. He was a legend.””Remembering the special times I spent with Mr Palmer at Bay Hill. A true pioneer for our sport. Forever remembered,” said Northern Ireland’s Rory McIlroy, hours after winning the Tour Championship in Atlanta.
“My heart aches with passing of the King. What he did for golf cannot be measured. Athlete, pioneer, philanthropist, family man, and much more…RIP Arnie,” tweeted US golfer Zach Johnson.
US President Barack Obama also paid tribute to “The King”, tweeting a picture of him visiting the White House.
The R&A, which runs The Open and jointly governs golf worldwide, called Palmer a gentleman who would live long in people’s memories.
Its chief executive Martin Slumbers said: “It is with great sadness that we have awoken to hear the news of Arnold Palmer’s passing. He was a true gentleman, one of the greatest ever to play the game, and a truly iconic figure in sport.
“His contribution to The Open Championship was, and remains, immeasurable. He will be missed and forever remembered by all at The R&A and throughout the world of golf as a charismatic and global champion of our game.”
Palmer was born in Latrobe, Pennsylvania, in 1929, the son of a groundskeeper at the local country club who later became the professional at the golf club there.
He was one of golf’s most dominant players in the late 1950s and early 1960s, winning seven major titles over seven seasons.
He also notched up 62 PGA Tour wins.
In a 1960 cover story written during his heyday, Time magazine swooned over Palmer, whose fans would flock to tournaments in droves:
“Win or lose, Palmer, with his daring, slashing attack, is fun to watch. He is a splendidly built athlete (5ft 11in, 177 lbs) with strength in all the right places: massive shoulders and arms, a waist hardly big enough to hold his trousers up, thick wrists, and leather-hard, outsized hands that can crumple a beer can as though it were tissue paper.
“Like baseball buffs, golf fans dote on the long-ball hitter; they pack six deep behind the tee to gasp in admiration as Powerman Palmer unwinds to send a 280-yard drive down the fairway.”
Palmer is survived by his wife Kathleen Kit Gawthrop, his daughters, stepchildren and a large extended family of grandchildren and great-grandchildren.