White passed away on Wednesday at the age of 80 following a short illness - just two days after the death of former vice-chairman Jack Gillespie.
White - who was manager of Rangers between 1967 and 1969 - remained a regular at the club's matches, including last season in the Third Division.
Chairman and fellow former boss Walter Smith told the club's website: "We are all saddened by the news of Davie's passing and our immediate thoughts are with his family.
"I remember when he was appointed as manager and he had such a difficult job at that time, not just because he was following Scot Symon, but because Celtic were enjoying the best period in their history under Jock Stein.
"Davie was also manager of Dundee when I was with Dundee United and, ironically, he brought the League Cup to Dens Park in 1973 by beating Celtic.
"He was a great Rangers man and scarcely missed a match at Ibrox."