The Portugese coach has been without a club since his unceremonious sacking as Man Utd boss back in December.
Neil Lennon has taken over the reins at Celtic Park until the end of the season, but it appears the club’s board has other ideas going forward.
Sutton said: “It’s so tough to stand here and say who will be the next Celtic manager because it’s a big job.
“There’s going to be a lot of managers in for it.
“Mourinho was heavily linked and I’m led to believe that there was something in that in terms of there being an offer.
“But why would he take the Celtic job? I just don’t get it.
“I don’t know the exact terms but there was an offer.”
Lennon has done an admirable job of steadying the ship since former Celtic boss Brendan Rodgers departed suddenly for Leicester .
The club now stands on the brink of an illustrious treble treble, with only a Scottish Cup final with Hearts to overcome.
Nevertheless, the proposition is not thought to be one that will entice Mourinho, who will have no shortage of options available to him.
Wherever his next move takes him, the 56-year-old made it clear he hopes to return to the dugout sooner rather than later.
Mourinho said previously: “I have a lot of qualities and one of them is to respect other clubs and their management.
“I have never spoken like ‘this club wanted me, they contacted me.’
“When I left Manchester United in December, I immediately took the decision I want to work from summer.
“I hope to be back to work in July in a project I really like.”