Goram revealed in an interview with the Daily Record in May that he had been diagnosed with terminal cancer.
He said he had been given about six months to live by doctors after declining the chance of chemotherapy, which might extend his life by an extra three months.
"Rangers Football Club are today deeply saddened to announce the death of our legendary goalkeeper, Andy Goram, following a short battle with cancer," a club statement said.
"In a 1999 supporters' poll, he was voted the Greatest Rangers Goalkeeper of all time, and after retirement, was still a regular face around Ibrox on matchdays and still wowed the crowd with some memorable appearances in legends teams.
"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."
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Bury-born Goram began his career at Oldham before moving north to play for Hibernian between 1987 and 1991.
Goram made 260 appearances for Rangers between 1991 and 1998, where he was simply known as 'The Goalie', winning five league titles, three Scottish Cups and two League Cups.
He won 43 caps for Scotland and was their first-choice goalkeeper at both Euro 92 and Euro 96.
After Rangers, he played for Notts County, Sheffield United, Motherwell, Manchester United, Hamilton, Coventry, Oldham, Queen of the South and Elgin.
Goram was part of the Manchester United squad that won the Premier League in 2001, making two appearances for the Red Devils.
At Queen of the South he became the first player to complete the set of Scottish medals when he claimed a Challenge Cup medal to go alongside winners' prizes in the Premiership, Scottish Cup and League Cup.
He also won international caps for Scotland as a cricketer.
Goram remains the only person to have played a first-class international match at cricket as well as a major international football match for Scotland.
Former United team-mate Gary Neville tweeted: "Terribly sad news about Andy Goram. RIP Andy. Love to all your family."