The Portuguese has been out of work since leaving a role at Old Trafford in December 2018.
Various landing spots have been mooted for his next job, with the 56-year-old revealing that he intends to fill another post this summer.
A shock switch to Scottish champions Celtic has generated plenty of speculation, with no decision taken at Parkhead regarding their next full-time coach.
Neil Lennon is overseeing the end to another domestic treble bid at Parkhead, but it remains to be seen whether he will be kept on for 2019-20.
The Hoops are certainly exploring other options, with Sutton claiming that Mourinho has been approached.
The ex-Celtic striker told BT Sport when quizzed on who he would like to see handed the reins: “[Massimiliano] Allegri from Juventus, [but] it’s not going to happen.
“It’s so tough to stand here and say who will be the next Celtic manager because it’s a big job.”
He added on links to Mourinho: “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.”
Sutton believes Celtic will move quickly to make an appointment once their 2018-19 campaign comes to a close.
Having got their hands on an eighth successive Scottish Premiership crown, a cup final date with Hearts is next up on Saturday.
“I think we’ll find out very quickly after the cup final,” said Sutton.
“It’s not just Neil Lennon, [former head of recruitment] Lee Congerton needs replacing.
“Celtic have a big job to do over the summer and when you go back and think about the mess they made last summer they need to step up.
“Rangers actually brought some pretty decent players this season so it’s going to be a bit of an arms race.
“Whoever gets the better of that may well decide who wins the league.”