Mulgrew, who is a very versatile footballer, has been on the books of Celtic since 2010 and has a contract with the Scottish Premiership giants until the end of the month, according to Transfermarkt.co.uk.
Celtic first-team coach John Kennedy recently told The Glasgow Evening Times that talks between the the club and the 30-year-old over a new deal are going on.
Brendan Rodgers, who was recently appointed the manager of the Bhoys following the departure of Ronny Deila, told The Telegraph earlier this week that he would like to keep Mulgrew in his team for the 2016-17 campaign.
Former Celtic midfielder and manager Hay has suggested that Rodgers should retain the services of Mulgrew.
“Charlie is a useful guy to have in your squad,” Hay wrote in The Glasgow Evening Times. “He has experience, he understands the club and his versatility can be significant, especially if you have a situation like this where you look very thin on the ground defensively.
“It goes without saying that the squad needs trimmed but Charlie is a guy that I think has a big role at the club and, personally speaking, I would be wanting to keep him.”
Mulgrew is 30 years of age, and can play as a centre-back, left-back and central midfielder.
From the recent comments of Kennedy and Rodgers, it is clear that Celtic want to keep the Scotland international for next season.