He was one of golf's dominant players in the 1950s and early 1960s, winning seven major titles over seven seasons.
Tiger Woods said: "Thanks Arnold for your friendship council and a lot of laughs." The US Golf Association called him "golf's greatest ambassador".
President Obama posted a photo of a lesson Palmer gave him in the Oval office at the White House.
Jack Nicklaus, whose sporting rivalry with Palmer spanned over half a century, 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 wrote on Twitter.
"Arnold transcended the game of golf. He was more than a golfer or even great golfer. He was an icon. He was a legend."
Nicklaus, 76, who won 18 majors to Palmer's seven, posted a series of snapshots on Instagram recalling their long friendship.
Northern Ireland golfer Rory McIlroy recalled meeting Palmer at his Bay Hill Club in Orlando, Florida - where a professional tournament named after him is held each March.
"Remembering the special times I spent with Mr Palmer at Bay Hill. A true pioneer for our sport. Forever remembered,"  McIlroy tweeted, 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.
Rickie Fowler, a member of the US Ryder Cup team, said Palmer's memory would not fade. "Legends never die ... you will live on forever Arnie ... thank you for being you and giving me the opportunity to do what I get to do every day!"
How The King changed the game
Palmer the commercial champion
Outside the golfing world, others too shared their farewells on social media.
President Obama paid tribute to "The King", highlighting Arnold Palmer's philanthropy.
Former US President George H W Bush, a keen golfer, said: "He brought golf to millions by his daring and caring. We miss him already."
Arnold Palmer was the first golf player to make $1m from playing the sport.
But he made much more than that from his many off-course endorsements, putting his name to a variety of products and services, from United Airlines to Cadillac cars.
Nowadays it is commonplace for sports stars to lend their names to commercial products.
But half a century ago such an association between sport and brands was ground-breaking.
Palmer, in association with marketer Mark McCormack, was the trailblazer.
It is a legacy for which today's high-earning stars, making astronomical sums from their own deals, should be eternally grateful.
Palmer was born in Latrobe, Pennsylvania, in 1929, the son of a groundskeeper at the local country club who later became a professional at the golf club there.
Apart from his seven majors, he also notched up 62 PGA Tour wins.
The much-loved veteran died at a hospital in Pittsburgh, Pennsylvania, where he was undergoing heart tests.
Tributes have flooded in for legendary US golfer Arnold Palmer, who died on Sunday aged 87.