Goram made 260 appearances for Rangers between 1991 and 1998, winning five league titles and three Scottish Cups.
He was capped 43 times by the national team and was the first-choice keeper at the European Championships in 1992 and 1996.
"The thoughts of the directors, management, players and staff are today with Andy's family, and would ask that their privacy is respected at this sad time," Rangers said in a statement.
Goram, who made his cancer diagnosis public in April, was a fan favourite at Ibrox and was voted by supporters the best Rangers goalkeeper of all time in a 1999 poll.
He later played for Sheffield United and Motherwell and had a brief loan spell at Manchester United in 2000-01.
Goram also represented Scotland in cricket four times between 1989 and 1991.