Goram, 58, who also represented Scotland, played for the Ibrox side between 1991 and 1998 - making 260 appearances over the period and winning five league titles, three Scottish Cups and two League Cups.
Born in Bury, Goram began his career at Oldham Athletic before moving north to play for Edinburgh side Hibernian, and then moving to Glasgow.
After his time at Rangers he played for a number of teams in both Scotland and England, including Notts County, Sheffield United, Motherwell for three seasons, Manchester United, Coventry, Queen of the South and then lastly Elgin City.
A statement on the Rangers FC website said the club was "deeply saddened to announce the death of our legendary goalkeeper Andy Goram following a short battle with cancer".
Paying tribute to Goram, the club said: "He is a member of the Scottish Football Hall of Fame and was named both the Scottish PFA and Football Writers' Player of the Year in 1992-93 - arguably the strongest campaign in Rangers' history.
"Goram also won international caps for Scotland as a cricketer and remains the only person to have played a first-class international match at cricket as well as a major international football match for Scotland."
It concluded: "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.
"Funeral details will be communicated in due course."