Mulgrew has been on the books of Celtic since 2010 and has a contract until June 2016.
Former Celtic midfielder and manager Hay believes that the Scottish Premiership giants should offer a new deal to the 29-year-old, who is a versatile footballer and can play as a centre-back, left-back or midfielder.
“I have to say that if it was down to me I would offer Charlie Mulgrew a new contract today,” Hay wrote in The Glasgow Evening Times.
“At 29 he is still a player who has plenty of years ahead of him in the game and it is the wealth of experience he has that I think makes him so valuable for Celtic,” added the former Scotland international.
Mulgrew is a very good player, although he has had injury problems in recent times. The Scotland international remains an important player for Celtic, and his performances have reported attracted interest from clubs in England.
According to The Scottish Sun, Premier League side Bournemouth and Championship outfit Blackburn Rovers are interested in signing the Scotsman in the summer transfer window.
Celtic are at the top of the Scottish Premiership table at the moment with 55 points from 24 matches. Ronny Deila’s side are level on points with second-placed Aberdeen and have a game in hand.