With Giampaolo Pazzini, Matias Silvestre, Daniele Bonera and M'Baye Niang already sidelined by long-term issues, the Italian club have suffered a spate of further injury problems.
Following Saturday's 2-2 draw with Torino, Allegri said: "Riccardo Montolivo, Mattia De Sciglio, Ignazio Abate and Stephan El Shaarawy will not play on Wednesday.
"We’ll make do with who we have."
Montolivo was withdrawn just before half-time as Milan struggled in Turin at the weekend, with the midfielder passing the captain's armband to the returning Kaka upon his departure.
However, his replacement Andrea Poli is reportedly also facing a battle to be fit in time for the visit of Neil Lennon's men to the San Siro.
Mattia De Sciglio, Ignazio Abate and El Shaarawy had already been sidelined ahead of the Serie A fixture against Torino, and have subsequently failed to recover in time to feature in Europe.
Milan and Celtic last met in the group stages of the 2007-08 Champions League, with the Italians suffering a 2-1 defeat in Scotland before running out 1-0 winners at home.
‘Even though it was late in the game, the Arsenal fans started singing my name. It felt like the moment I announced myself as a professional footballer’: Ex-Gunners star recalls scoring first senior goal as a teenager at Highbury
'I started to drive myself crazy': Former Manchester United midfielder explains why it all went wrong for him under Erik ten Hag
