Mourinho was sacked by Man United in December and hasn’t managed any club since then.
The 56-year-old had reiterated his desire to return to management this summer and hinted that it would be in a league that motivates him.
But Sutton has now disclosed that the Portuguese gaffer was presented with the opportunity of coaching the Scottish giants next season.
“There’s going to be a lot of managers in for it (the Celtic manager’s job),” Sutton was quoted by Manchester Evening news as saying.
“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.”