Gill spoke to Sky Sports News after it was confirmed that 39-year-old Giggs had agreed a new one-year deal at Old Trafford.
The United chief was full of praise for Giggs and admitted that when he did hang up his boots, he would have an immediate role at the club offered to him.
On his future Gill said: "We will see, but he will have a great role to play at Manchester United after he finishes his playing career.
"He is an embodiment of what United is all about, we gave him a chance in the 80s early 90s and he is embodiment of professionalism, he is about caring, always wanting to improve and he has changed in that period and that is why he still playing at the very top of club football.
"His desire is still there, his motivation is still there and as Alex says the ability is still there, so let's hope that continues for a few years to come."
Gill refused to be drawn on what type of role Giggs could end up in at United.
He added: "We have not had a major discussion, Ryan is doing his pro licence and is preparing for that so we will have to see what he wants to do, coaching or managing, but he has been one, if not the most successful players at United and we will have to wait and see."
Gill hailed Giggs' stay at United, saying: "It is just a testament to Ryan Giggs as a footballer and a person, so stay on top of your game for 23 years scoring every season is brilliant for the club and himself, so it was a very easy decision, Alex spoke to him and gave him another year and it was done very quickly."
He says that Giggs' health was never an issue, adding: "We always speak to doctors before we offer a player a new contract and they said he is as fit as ever and in great shape as an athlete.
"You saw that last weekend running the show against QPR with Michael Carrick and that is testament to him, the medical team at Old Trafford and what he has achieved on the training pitch and on the pitch itself."
Giggs is to play his 1000th senior game this weekend and is on course to add a 13th league championship medal to his collection with United 12 points clear of closest rivals Manchester City.
The former Wales international is set to celebrate his milestone in front of a home crowd against Norwich and boss Sir Alex Ferguson says he expects the evergreen winger will remain in a coaching capacity once he does decide to end his playing days.
"He'll be involved (against Norwich) and he'll be involved on Tuesday (against Real Madrid) so somewhere along the line he's going to play his 1,000th game this week, which is phenomenal," Ferguson told United's official website.
"What can I say about Ryan that hasn't already been said? He is a marvellous player and an exceptional human being.
"He has fantastic energy for the game and it is wonderful to see. To think that he now has 23 unbroken years of league goals behind him is truly amazing in the modern-day game.
"His form this year shows his ability and his enjoyment of the game are as strong as ever and I am absolutely delighted that he has signed a new contract.
"(A coaching role) will happen, that's the plan. We've done the same with Nicky Butt with the reserves and when Paul Scholes finishes it will be exactly the same with him. I think it's important we have people here who have the experience of being a player here."
Giggs revealed his delight, saying: "I am feeling good, enjoying my football more than ever and, most importantly, I feel I am making a contribution to the team.
"This is an exciting team to be part of, with great team spirit, and we are again pushing for trophies as we head towards the business end of the season.
"I am delighted to have signed a new contract. I would like to thank Sir Alex and the coaching staff and I would also like to thank (outgoing chief executive) David Gill who has been a tremendous support over my many years here."