Smith scored 208 goals in 317 senior games for Spurs and was a member of the side which won the double in the 1960/61 season.
He also had spells at Chelsea and Brighton.
"It is with great regret that we announce the death of Bobby Smith, our legendary centre-forward from our great double-winning side of 1960-61," read a statement on Tottenham's official website.
"Bobby died at an Enfield hospital yesterday evening [Saturday] following a short illness.
"We extend our sincere condolences to the family of Bobby Smith at this very sad time."
Smith began his career as a youth player at Chelsea and turned professional in 1950.
He moved to Spurs in December 1955 and played a key role in the club's transformation from relegation candidates to champions.
He scored 33 goals in the club's double-winning season and picked up winner's medals in the 1962 FA Cup final and the 1963 European Cup Winners' Cup.
Smith was also prolific at international level, scoring 13 times in 15 appearances for England.
Source: PA