Players will wear black armbands at their next league game as a tribute to season ticket holder Robert Learmonth.
The 57-year-old supporter became unwell at Ibrox stadium during his club's game against Berwick Rangers on Saturday.
Emergency services helped Mr Learmonth's at his seat in the Broomloan Stand in the first half of the match. He was taken to the nearby Southern General Hospital but later died.
The club has sent condolences to Mr Learmonth's family who also lost Robert's dad earlier that same week.
Rangers chief executive Charles Green said: "On behalf of everyone at Rangers Football Club I would like to pass on my sincere condolences to the family and friends of Robert Learmonth.
"Robert was a season ticket holder and a lifelong supporter and we were all shocked and saddened to hear the sad news he had passed away at the stadium on Saturday.
"It is unthinkable that anyone can come to a football match and not return home and our thoughts are with Robert's family and friends at this difficult time."
Team manager Ally McCoist said: "It was devastating to learn a supporter had passed away at the stadium on Saturday and the tragic news certainly puts football into perspective.
"The Learmonth family were already mourning the passing of Robert's father who tragically died last week, so as a mark of respect the players will wear black armbands at Sunday's Third Division match against Peterhead at Balmoor Stadium."