Tommy Craig, who was with the Ibrox club for around 12 years from the early 1970s, is believed to have passed away in hospital on Saturday.
He was on the staff when the team won the 1972 European Cup Winners Cup against Dynamo Moscow, beating them 3-2 at the Camp Nou in Barcelona.
After leaving the club Mr Craig practised physiotherapy from his home in the South Side of Glasgow.
Mr Craig was taken to hospital after suffering a heart attack but did not recover. He is thought to have been in his early 70s.
Derek Johnstone said: "Tommy was at the club for 12-13 years when I was a kid.
"He was a great physio. Players were always back up on the pitch earlier than expected.
"He would have worked with Barcelona too, during the Cup Winners Cup match."