Deila has been in charge of Celtic since the summer of 2014 and has a contract until the end of the season.
The Hoops are on course to winning the Premiership title once again this campaign, but Deila has come under pressure.
Celtic have failed to qualify for the group stage of the Champions League for the past two seasons and will not feature in the knockout rounds of the Europa League this campaign.
Stewart, who is a well-known Celtic fan, has suggested that Deila should be dismissed and replaced by David Moyes.
“If I was to say anything about the manager, I just think that maybe the club’s a bit too big for him at this point in his career,” the 70-year-old told talkSPORT. “I don’t want to run him down and mock him but that’s the way I look at it.”
“I can’t say that, I’m very close to the board there and the executive chairman,” added Stewart when asked whether former Manchester United and Everton boss Moyes would be a good fit at the Scottish champions. “Something is wrong, but I wouldn’t mind David Moyes at all!”
Moyes is a very good manager who did a great job at Everton from 2002 to 2013.
Despite his failures at United and Real Sociedad, the Scotsman remains someone who should be given another chance at a top club.
Moyes is without a managerial job at the moment, having been sacked by Spanish club Sociedad in November.