Gilzean was capped 22 times for Scotland, and was diagnosed with a brain tumour just weeks ago.
The striker, known as the 'King of White Hart Lane,' signed for Tottenham in 1964, scoring 133 goals over his 10-year tenure.
Before that, he has spent seven years at Dundee, winning the league title in 1961-62 and helping the team reach the European Cup semis the following year.
While at Tottenham, he won the FA Cup, two League Cups and was part of the team that won the all-English UEFA Cup against Wolves in 1973.
Tottenham released a statement which read: "Everyone at the club is deeply saddened to learn of the passing of our legendary former striker Alan Gilzean."
"Our thoughts are with his family and friends at this extremely difficult time."
Dundee also released a statement with thoughts for Gilzean's friends and family:
"Everyone at Dundee Football Club is devastated to learn that club legend Alan Gilzean passed away this morning after recently being diagnosed with a brain tumour. The thoughts of everyone at the club are with Alan's family and friends at this difficult time."
Gilzean was a popular man at his teams, and teammate Jimmy Reaves tweeted his condolences, remembering his colleague:
July 8, 2018