Hay has been told his 18-year playing career at Rugby Park is over, despite being promised a new player-coach contract by subsequently sacked manager Kenny Shiels before the end of last season.
But former Rangers player Kyle feels Hay's experience is invaluable and has urged Rugby Park chairman Michael Johnston to have a re-think.
Kyle, who was today at Ibrox to launch a new book by Lisa Gray, 'Rangers FC: We Don't Do Walking Away', said: "It's very rare in this day and age that you hear someone serve 18 years at one football club. I think only Ryan Giggs has probably beaten that.
"It's a sad day for Garry but he knew this time would come eventually.
"But I'm sure somewhere along the line, they will find a role for him at Kilmarnock.
"He knows Kilmarnock inside out and he has been a great servant.
"Obviously winning the League Cup last year would have been one of his proudest moments and he was there when they won the Scottish Cup as well.
"He was a great guy around the dressing room with a lot of experience and a lot of good advice to younger guys.
"I know Garry has been doing a lot of coaching badges so hopefully there will be a role for him somewhere and he can continue to do what he has done at Kilmarnock all those years."
- Sports & Recreation
- Kenny Shiels