The 79-year-old played more than 300 games for Spurs between 1964 and 1974, winning numerous trophies during his time at the club including the FA Cup, the UEFA Cup and the League Cup.
Gilzean is also revered at Scottish club Dundee, where he scored 169 goals in 190 appearances between 1957 and 1964 ahead of his move to White Hart Lane.
"Our thoughts are with Alan Gilzean and his family after our legendary striker was recently diagnosed with a brain tumour," Tottenham wrote on their Twitter account.
"The club and Tribute trust are in contact with Alan's family, who have asked for privacy at this time. We send them all our best wishes."
Gilzean made 22 appearances for Scotland, scoring 12 goals, and was inducted into the Scottish Football Hall of Fame in 2009.