Giggs is now 37 and will soon have completed 20 years with the club and his current contract is almost up.
But Gill is making no secret of the fact that United would like to keep him for at least another year.
He told the Daily Telegraph: "I wouldn`t be surprised if he extends his contract for another season.
"That will be addressed in the next month or so. He has been playing well, keeps himself in great shape and is a wonderful advert for the modern footballer."