The 39-year-old, who is closing in on his 1,000th senior appearance, signed a one-year contract extension at Old Trafford on Friday.
"We will see, but he will have a great role to play at Manchester United after he finishes his playing career," Gill told Sky Sports News.
"He is an embodiment of what United is all about. We gave him a chance in the '80s and early '90s and he is embodiment of professionalism."
Gill recently announced that he would be stepping down from his position as chief executive at the end of the season.