Charlton was a key member of England's victorious 1966 World Cup team and also enjoyed great success with United, who became the first English club to win the European Cup in 1968, a decade after members of the team were killed in an air crash in Munich.
"It is with great sadness that we share the news that Sir Bobby passed peacefully in the early hours of Saturday morning. He was surrounded by his family," said a statement on behalf of the Charlton family.
In 2020, it was announced Charlton had been diagnosed with dementia and as the disease took hold he stopped attending matches at Old Trafford.
United posted a picture of Charlton on X, formerly known as Twitter, with the caption: "Words will never be enough."
United players and their opponents from Sheffield United paid tribute to Charlton before their Premier League clash at Bramall Lane.
United captain Bruno Fernandes laid a wreath and a picture of Charlton was shown on a giant screen as the crowd applauded.
Wayne Rooney, who beat Charlton's scoring record at United, said he was in shock at the death.
"Sir Bobby was always great with me. We had conversations not just about football but about life," said Rooney, now the manager of Birmingham City.
"He's a huge loss to football and a huge loss to his family, more importantly. He'll be greatly missed."
Charlton and the manager of the team, Matt Busby, survived the 1958 Munich air disaster which claimed the lives of eight of Charlton's team-mates.
He recovered from his own injuries and returned to the shellshocked team.
A decade after the crash, with a team that by then included the prodigiously talented George Best, Charlton scored twice as United beat Portuguese side Benfica 4-1 following extra-time in the European Cup final at Wembley.
Two years earlier, the midfielder played a leading role in England's World Cup win alongside his brother Jack, who died aged 85 in 2020. Bobby won 106 caps for England, scoring 49 goals.
He made his debut for United in 1956 and went on to play 758 matches for the Red Devils, scoring 249 goals. Both were long-standing club records until they were overtaken by Ryan Giggs and Rooney respectively.
Charlton won three league three titles and one FA Cup at Old Trafford and, after leaving United in 1973 and becoming Preston manager, he returned to Old Trafford 11 years later as a club director. He was knighted for services to football in 1994.
"Manchester United are in mourning following the passing of Sir Bobby Charlton, one of the greatest and most beloved players in the history of our club," the club said in a statement.
"Sir Bobby was a hero to millions, not just in Manchester, or the United Kingdom, but wherever football is played around the world.
"He was admired as much for his sportsmanship and integrity as he was for his outstanding qualities as a footballer. Sir Bobby will always be remembered as a giant of the game."
FIFA president Gianni Infantino said Charlton's influence on football had been long-lasting.
"We mourn the loss of one of England's 1966 FIFA World Cup winning team and a football legend, whose impact on the game spanned generations," Infantino said.
Gary Lineker, the former England forward and now a BBC presenter, said Charlton was "for me, England's greatest ever player".
"He may no longer be with us but he'll have footballing immortality. RIP Sir Bobby."
Prince William, who is president of the Football Association, described Charlton as "a true great who will be remembered forever" in a message on social media.
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