Mr Redford, 53, was found dead in Irvine, North Ayrshire, at about 11.15am yesterday. There are no suspicious circumstances.
The retired midfielder, who lived in Saltcoats with his wife Janine, became a major star after joining Rangers from Dundee for a then Scottish record fee of £210,000 in 1980.
Mr Redford's exploits included scoring the winner in Rangers' 1981 League Cup final victory.
He won four domestic trophies before signing for Dundee United in 1985 and going on to play a pivotal part in their march to the Uefa final against IFK Gothenburg in 1987, which they lost 1-0.
Mr Redford scored one of the club's two goals against Borussia Monchengladbach in the semi-final.
Former team-mates last night spoke of a 'nice guy' who remained in touch through the club's regular golf outings. Tommy McLean, a former Rangers team-mate said: "He was a really lovely lad. A great lad off the park and mixed well with the players. He did well with Rangers during his time there and he was a good team-mate. It's his family I'm thinking about."
John MacDonald, who played with Redford at Rangers in the 1980s, said: "He was great to play with, I remember him scoring the winner in the 1981 League Cup final against Dundee United and that was a great day.
"You couldn't meet a nicer guy off the park. I met him at a few Rangers golf days over the years and he was great, even though we'd all slowed down a bit because we were all older. It's terrible to hear, really unbelievable."
Rangers manager Ally McCoist paid tribute to Redford, who he played with in the 1980s.
There will be a minute's silence held in memory of the 53-year-old before Rangers' match against East Fife this afternoon.
McCoist said: "We are deeply shocked to hear the very sad news that Ian has passed away. I had the honour of playing with him early in my Rangers career and he was a terrific lad and an extremely talented footballer who will be sadly missed.
"Everyone at Rangers sends their sincere condolences to Ian's family and our thoughts are with them at this very difficult time."
In a statement, Dundee United said: "Everybody at Dundee United is devastated to hear of the passing of ex-United player Ian Redford. The memory of his performances for United at a time when the club competed in the higher echelons of European football will forever be part of our history.
"Our thoughts and prayers are with Ian's family at this very sad time." A Dundee spokesman said: "Everyone at Dundee FC was saddened to hear of the death of former midfielder/striker Ian Redford, one of the most talented kids to come through the Dens youth system over the last few decades."
Former Scotland player Kevin Gallacher said on Twitter: "just heard news about Ian Redford, thoughts go out to his family". Conservative MSP Murdo Fraser also wrote on the social networking site: "Very sad to hear of the untimely death of Ian Redford. A great servant to the club #RFC"
The Perth-born player's early life was marred by tragedy when his seven-year-old brother Douglas died after a long battle with leukemia when the future footballer was 12. Mr Redford said he often fought with him and teased him, as his parents didn't tell him or his sister Jill of the gravity of his illness. He said the tragedy left him angry and that football was his escape.
After six seasons at Rangers, where he made almost 250 appearances, he joined Jim McLean's team in 1985. He won a runners-up Uefa Cup medal and Scottish Cup medal in 1987.
He then moved on after three years to Ipswich Town, before playing for his home town club, St Johnstone, and Raith Rovers, where he won a League Cup medal after beating Celtic in 1995.
The player suffered from deafness in his one ear from a young age, and began losing hearing in the other aged 40. A Police Scotland spokeswoman said: "Officers are investigating and inquiries are continuing. A post mortem will be carried out in due course to establish the cause of death."