The former Celtic striker has asked whether Deila is the right manager to lead the Bhoys to success in Europe.
Hartson has also questioned why Celtic have been defensively vulnerable so far this season.
“If the Celtic board want someone merely to keep on winning the Scottish Premiership then Deila’s the man,” the former Wales international wrote in The Scottish Sun.
“If, however, they are seeking someone to take the club forward in Europe then Deila should be nowhere near it. What’s it to be?
“Peter Lawwell and the directors have some serious thinking to do ahead of Celtic’s final two Europa League ties with, incredibly, qualification for the last 32 still there for them.
“But if I was on the board I’d be demanding answers from Deila as to why, in particular, the defence is consistently abysmal.”
Ronny Deila under pressure
While Celtic look set to win the Scottish Premiership yet again, their chances of progressing to the knockout stages of the Europa League look slim.
The Hoops went down 2-1 at home to Molde in the Europa League Group A on Thursday evening.
Deila’s side’s defeat at Celtic Park means that they are now at the bottom of the group with just two points from four matches, three points behind second-place Fenerbahce.
The Celtic boss has come under pressure, but there is still a chance that he can guide the team to the knockout stages.