The Scottish club's manager, Ronny Deila, has ruled out recruiting the 29-year-old winger this month, telling The Herald that wage demands pose a major problem.
Former Celtic striker Sutton believes that the Hoops should do all they can to recruit the Republic of Ireland international and bring him back to Celtic Park.
“Celtic need to show the fans they are ambitious and bringing someone like Aiden back would do that,” the Englishman has told The Daily Record.
“It would send a message that they are trying to add experience and character, something which Ronny says this group needs. McGeady has been there, done it, knows the club inside out and needs game time ahead of the Euros.
"I know there are financial difficulties but this deal can work if the clubs agree a percentage of the wages each.”
McGeady started his professional club career at Celtic and played for the Scottish giants until 2010.
The winger then moved to Russian Premier League club Spartak Moscow and joined Everton in January 2014.
Things have not gone according to plan at the Toffees for the Irishman, who is struggling for playing time at the moment.
McGeady has not made a single Premier League appearance for Everton so far this season, while last season he scored one goal and created 17 chances in 16 league games.