The former Ferencvaros forward, who turned 70 last month, won Europe's top individual award in 1967 and made 75 appearances for his country, scoring 31 goals.
He died following complications after heart surgery last week, the FA reported.
Born in the village of Hercegszanto in southern Hungary in September 1941, he became known as "The Emperor" in his homeland, and stayed a one-club player for his whole career.
He made his debut for Ferencvaros as a 17-year-old in 1958 and in all he made 537 appearances, scoring 383 goals before retiring in 1974.
Despite the brilliance of the Hungary's Magical Magyars team of the early 1950s, the European award, which began in 1956, passed them by and Albert remains the only Hungarian to win the honour.
"You know what is so special about the Golden Ball - the defenders never took it away from me," he told Nemzeti Sport, Hungary's daily sports newspaper on his 70th birthday last month.
As well as winning four Hungarian League titles with Ferencvaros, he was also part of the Ferencvaros team which won the 1965 Inter Cities Fairs Cup, precursor of the UEFA Cup, beating Juventus in the final in Turin.
A tall and elegant player, with excellent close control, Albert was joint top scorer with four goals in the 1962 World Cup finals along with five other players including Garrincha and Vava of Brazil. He was also named Young Player of the tournament.
Albert's son, Florian Albert Jr, also played for Ferencvaros and Hungary and the club's stadium is named after him.
There have also been concerns recently regarding another Hungarian great, Gyula Grosics, the goalkeeper of the Mighty Magyars team of the 1950s.
He is recovering in hospital following a fall, but was not in any immediate danger, a hospital spokesman said at the weekend.
