The result could also have a bearing on the future of Celtic manager Ronny Deila as the loser from the match will be knocked out of the tournament should Fenerbahce beat Molde, who are already through to the round of 32.
Celtic supporters are used to watching their team on European nights among the elite in the Champions League, but in the last couple of seasons, the club have failed to qualify for the group stages.
Therefore, the Scottish giants have been playing in the Europa League, but this campaign under Deila has been less than fruitful as the club are in danger of failing to reach the round of 32.
The Bhoys have not won any of their four games in the group, recording two draws and two defeats, the latest of which came in the form of a 2-1 loss to Molde, who have already qualified for the next phase of the competition.
Deila may have overseen just one defeat in 15 Scottish Premiership games this season, but there are still question marks as the team were frustrated in a goalless draw against Kilmarnock at the weekend.
The Norwegian manager has put a lot of emphasis on playing the 'Celtic way' - an entertaining style of football, but it has resulted in the team's defence struggling as they have conceded 11 goals in the league and nine in four Europa League games.
This Thursday's opponents will provide a tough test for Celtic as they managed to come from behind twice to seal a 1-1 draw on matchday one in Amsterdam.
The Europa League has not been a happy competition for Celtic in recent seasons as they have not managed to claim a win in nine consecutive matches.
Recent form: DDLL
Recent form (all competitions): WWWLWD
Frank de Boer's side sit one point above bottom-placed Celtic in Group A, so have a better chance of snatching the second qualification spot, but they will need to gain all three points at Parkhead in order to help their chances.
The campaign so far has not been what they desired as they have failed to earn a victory in the group after progressing via the qualifiers following a two-legged tie against FK Baumit Jablonec.
It was a positive start for Ajax as they fought back to claim a point from their opening group match against Celtic, and they had a similar result when they came from behind to snatch a 1-1 draw away to Molde.
However, since then, the Dutch team have lost 1-0 to Fenerbahce and failed to strike in a stalemate at home to the Turkish club.
The statistics are not great reading for Ajax as they have not won in five continental away matches having drawn three and lost two, but they have progressed into the round of 32 in all three of their previous campaigns in the UEFA Cup and Europa League.
Another positive note for Ajax is that they are flying high in the Eredivisie as they currently sit at the top of the league table and are unbeaten in their last five games.
Recent form: WDDDLD
Recent form (all competitions): WLWDDW
Celtic captain Scott Brown has been ruled out of Thursday's fixture as he faces a possible three-month layoff with a knee injury.
The 30-year-old midfielder, who has played for the Scottish team for eight years, missed four matches before returning to the weekend's lineup against Kilmarnock.
However, the skipper has been dealt a blow, while Kris Commons will also miss the match as he deals with illness and a muscle injury.
Carlton Cole's fitness is not believed to be up to scratch as yet after joining the club on a free transfer, so the striker is likely to sit this game out, meaning that Leigh Griffiths will lead the line once again.
Jozo Simunovic is likely to remain at the back in a bid to stop the defence from leaking goals, while Izaguirre, who was sent off last time out against Ajax, should start the game.
Celtic possible starting lineup:
Gordon; Lustig, Boyata, Simunovic, Izaguirre; Allan, McGregor; Forrest, Rogic, Armstrong; Griffiths
Ajax possible starting lineup:
Cillessen; Tete, Veltman, Riedewald, Dijks; Klaassen, Gudelj, Serero; Younes, Milik, Fischer
Head To Head
The two sides have played one another nine times in three different competitions - the European Champions Clubs' Cup, the Champions League and Europa League.
Prior to September's 2-2 draw in Amsterdam, the last time that the two clubs met was in 2013 when they gained a win apiece in the group stage of the campaign.
Ajax earned a 1-0 win on home turf, while Celtic claimed a 2-1 triumph in Glasgow, but overall, the Scottish club edge the head-to-head record with four wins compared to the Dutch club's three.
Four of the nine matches played have ended in draws, while both sides have scored 12 goals each during their meetings.
We say: Celtic 1-2 Ajax
Thursday's match could spell the end of Celtic's Europa League campaign as Ajax have the quality to upset the Scottish side at Parkhead. It has not been happening for Deila in Europe this season, and the team are allowing too many goals to get shipped past them. The crowd will be up for the encounter, but if Deila is not prepared to change tactics in order to get the best result, then a defeat could be on the cards.