The Welshman signed a one-year contract extension on Friday, which will keep him at the club beyond his 40th birthday.
Giggs became the only player to score in each of the 21 seasons of the Premier League when he notched his first goal of the season at home to Everton last month.
"We will see, but he will have a great role to play at Manchester United after he finishes his playing career," outgoing chief executive David Gill told Sky Sports.
"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.
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"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.
"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 they most successful players at United and we will have to wait and see.
"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.
"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."