Brutally beaten 4-1 home and away by Sparta Prague, Neil Lennon's team are out of the running after just four matches, while Milan's failure to beat Lille in back-to-back fixtures means that they have surrendered the Group H top spot to the French side.
Stefano Pioli's men are now just one point above Celtic's conquerors Sparta in second place but can still secure qualification as early as this week if they turn over a reeling Celtic side and the Czech club lose in Northern France.
Hauge's fellow Norway international Mohamed Elyounoussi's strike for the home side proved in vain and their continental campaign has been on largely downward spiral ever since.
Milan start as particularly strong favourites this Thursday, as the Rossoneri's home record in the Europa League is W9 D1 L3. By comparison with their visitors' woeful results on the road, that makes the odds of an upset at San Siro almost insurmountable.
Celtic failed to negotiate the Champions League qualifying phase for the third successive summer, though this is the tenth successive season that the club have competed in a European group stage.
The Bhoys have occasionally made it into the last 32 before, though never further. This term, even that minor measure of success will not be achieved.
Horrendous away form in Europe has long been a burden on their ambitions. Celtic's win at Lazio last season was only their second ever victory outside Glasgow in the Europa League proper. Their all-time away record in the competition now reads W2 D11 L11, following this season's 2-2 draw in Lille and subsequent defeat in Prague.
Celtic are increasingly losing touch with their bitter rivals Rangers at the top of the SPL - though with games in hand - and the current points gap is wider than most fans can tolerate. At the weekend, the Hoops crashed to a disastrous cup exit against Ross County when, despite having three-quarters of the possession, they went down 2-0 at home.
All of which has heaped more pressure on manager Neil Lennon, who has faced - and vociferously rejected - a stream of criticism from fans and pundits alike. Thursday's glamour trip, then, may be the last thing the former Leicester City midfielder needs right now, as another European thumping can only embolden his critics and sharpen knives in the boardroom.
These clubs are familiar foes having faced each other ten times prior to this campaign, all in the European Cup or Champions League. Unsurprisingly, Milan recorded six victories to Celtic's one in those encounters, with four wins and a draw at home.
On Sunday afternoon, the Rossoneri - still adjusting to life without talisman Zlatan Ibrahimovic since his recent injury - took on Fiorentina and recorded yet another Serie A win by virtue of two first-half goals, once more with Ante Rebic leading the line in place of 'King Zlatan'.
The Swedish star aside, it is a young Milan squad with apparently no room for Uruguayan left-back Diego Laxalt - squeezed out by the electric form of Theo Hernandez last term - who joined Celtic on loan from Milan in September and has been a lively presence for Lennon's toiling team.
The likelihood of this return to his parent club resulting in anything other than a defeat is slim though, as the Serie A leaders will be focused on securing European football into 2021 and are unlikely to slip up now.
AC Milan Europa League form: WWLD
AC Milan form (all competitions): WLDWDW
Celtic Europa League form: LDLL
Celtic form (all competitions): WLWDLL
AC Milan are still without Stefano Pioli back ahead of Thursday night's game - the former Fiorentina manager tested positive for COVID-19 over two weeks ago - so Daniele Bonera stays in the dugout. Pioli has generally selected a much-rotated XI in Europe and will do so again on Thursday.
Key man Zlatan Ibrahimovic is unlikely to return until next week, while Samu Castillejo and Rafael Leao are also injured and have been ruled out. However, Belgium winger Alexis Saelemaekers has recovered and Ismael Bennacer is back in training but may not be involved.
For under-pressure Neil Lennon, Celtic star James Forrest is still out after ankle surgery and will not return for several weeks yet. The beleaguered boss switched to a three-man defence in the humbling against Ross County but will more likely use a 4-2-3-1, with Odsonne Edouard at its apex.
AC Milan possible starting lineup:
Donnarumma; Hernandez, Romagnoli, Gabbia, Dalot; Kessie, Tonali; Saelemaekers, Krunic, Hauge; Rebic
Celtic possible starting lineup:
Bain; Frimpong, Ajer, Jullien, Laxalt; Brown, McGregor; Ntcham, Christie, Elyounoussi; Edouard
We say: AC Milan 3-0 Celtic
Celtic's crisis of confidence deepens by the day while their hosts are now officially the most in-form team in the top five European leagues - with a greater points average than Bayern Munich or Premier League leaders Tottenham Hotspur.
Milan - even without the influence of Ibra - can take care of business at San Siro, as their youthful 'shadow side' are more than capable of shredding the Bhoys' dodgy defence.