The 30-year-old is currently out of contract with the Scottish Premiership side.
Celtic's Charlie Mulgrew
The Scotland international is out of contract at Parkhead this summer (via Sky Sports), but former Hoops striker Sutton is in no doubt that Mulgrew still has plenty to offer Brendan Rodgers' side, and outlined just how important the utility man is to the club.
"There is also the Charlie Mulgrew situation which needs resolving quickly. He has a future at the club. He's versatile and can play a number of positions well," he told the Record.
"Players like Charlie are a dream for a manager because of their reliability to step in at short notice in numerous slots and do a decent job time and again.
"But it seems to be a stand-off at present with Charlie waiting and assessing his options but Brendan saying: 'I like you but on my terms'."
Celtic manager Brendan Rodgers
As Sutton suggests, Mulgrew's presence alone gives the Bhoys' squad added depth in several different positions in both defence and midfield.
While he plays every role asked of him to such a standard that it is difficult to know which is actually his best position, it was his performances at centre-back last season that were so important to the club clinching their fifth successive league title.
His partnership with Erik Sviatchenko brought some stability to an area of the pitch that had previously looked to be Celtic's weakness during the time that Mulgrew was sidelined with injury and before the Dane arrived at the club.
And with Rodgers yet to strengthen his options in that department, and with the beginning of the Champions League qualifiers just around the corner, it would arguably be a major gamble for the manager to let the long-serving stalwart move on.
Mulgrew's departure would leave a considerable void in Celtic's squad, and there is no guarantee that any new signing replacing him will be capable of filling it with the kind of reliability that the current vice-captain brings.