The Dutch champions have amassed one point from their opening three group games, courtesy of a home draw with Milan at the start of last month.
Never before have Ajax failed to win in their first three fixtures of the competition, although a victory at Amsterdam ArenA against Celtic will lift them above the Scottish club in Group H.
Frank de Boer's side do not head into the match in particularly impressive form, their only win in five outings coming against fourth-tier ASWH in the KNVB Beker.
Celtic defender Efe Ambrose anticipates a tough game in the Netherlands, but he believes all the pressure is on Ajax as the hosts look to move off the bottom of the group and avenge a 2-1 defeat when the two teams met in Glasgow a fortnight ago.
Ambrose said: "Going to Amsterdam is huge for us.
"It'll be a deciding game for us – every one we play from now on is crucial, a cup final, to make the last 16. We know what we're up against and we know we have to get everything right.
"It's important to believe we can do it and we're working hard to achieve our goals. We know we can't write Ajax off though. They played good football against us at home, but they didn't take their chances. We took ours.
"They're a good side but the most important thing is for us to keep our heads straight and stay focused because we know if we lose it's all over. We don't want to let that happen.
"We're comfortable going there now, even though it won't be easy. But they know they can't lose the game if they have any chance to qualify. The better team will win the game, but I believe we're the better team."
Neil Lennon's Celtic side have lost their last two European away matches, in the play-offs at Shakhter Karagandy and against Milan on matchday one, but the Scottish Premiership club did emerge 3-1 victors on their last visit to Amsterdam in 2001.
A 1-1 draw at home to Dundee United on Saturday saw Celtic extend their unbeaten start to the league season to 11 matches, their longest streak since 2003-04, while Ajax lost 1-0 to Vitesse Arnhem.
Lennon will be without the services of Adam Matthews (broken collarbone) and Steve Mouyokolo (Achilles), but Georgios Samaras, Kris Commons, Mikael Lustig, Teemu Pukki and Beram Kayal could all return to the starting line-up. Nir Biton is suspended following his red card against Ajax last month.
Bojan Krkic (hamstring), Niklas Moisander (knee) and Lerin Duarte (ankle) are expected to be unavailable for Ajax.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.