Deila has come under pressure due to the Bhoys’ disappointing performances and results in Europe.
The Scottish Premiership champions went down 2-1 to Norwegian side Molde at home in the Europa League Group A on Thursday evening.
The defeat at Celtic Park means that Celtic are now at the bottom of the group with just two points from four matches, three points behind second-place Fenerbahce.
Former Celtic striker Sutton believes that Deila’s future as Bhoys manager is under threat and has written in The Daily Record: “You have to fear for Deila. I don’t know what the club are thinking but Ajax is looking like a last stand for the manager.
“The games against Ajax and perhaps Fenerbahce will decide Ronny’s fate. The performances will be equally as critical as the results.
“Lose in a limp manner in either game and Peter Lawell will be left with little choice – however reluctant he may feel to sack his man in the dugout.”
Celtic will take on Dutch giants Ajax at Celtic Park in Glasgow later this month and face Turkish giants Fenerbahce away in Istanbul in December.
The Hoops have not qualified for the Champions League for two successive seasons under Deila, and failure to progress to the knockout stages of the Europa League will not go down well with the Celtic board.