The 37-year-old Welshman, who is closing in on 20 years as first-team player at Old Trafford, is approaching the end of his existing deal.
Carry on: Manchester United are keen for veteran midfielder Giggs to extend his contract beyond the end of the season
But Gill is keen to keep him with the club for another season.
'I wouldn`t be surprised if he extends his contract for another season,' he told the Daily Telegraph.
'That will be addressed in the next month or so.
Centre of attention: Giggs made his 600th club appearance in the league against Tottenham last weekend
'He has been playing well, keeps himself in great shape and is a wonderful advert for the modern footballer.'
United manager Sir Alex Ferguson called Giggs an 'incredible human being' on Sunday after he made his 600th league appearance for the club against Tottenham.
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