The Italian insists his side must improve upon their display in Saturday's 2-2 draw with Torino, despite a string of injury problems which "read like a wartime news bulletin"
Mario Balotelli's 96th-minute penalty salvaged a 2-2 draw on Sunday after the hosts had led 2-0 until the 88th minute of the match, and Allegri has warned that the Scottish champions could pose an even sterner test in their Champions League Group H opener at San Siro.
"We played very badly against a Torino side that caused us a lot of problems [on Sunday]," Allegri told a press conference. "Tomorrow will be a different game as they always are in the Champions League. It’s a great atmosphere.
"It’s a big match as we’ll be up against a team that are very physically strong as well as being technically good players, especially in the attacking department. They’ll be a tough team to face.
"The Champions League is different to Serie A. We can’t give away as much as we did with Torino. If you defend better, you attack better. It won’t be easy against Celtic and Ajax as it’s never easy in Europe. I think that Celtic are a strong team and we’ll see what happens."
The Rossoneri have been blighted by injuries in the opening few weeks of the season, with Kaka the latest to join the likes of Giampaolo Pazzini, Daniele Bonera, M'Baye Niang, Riccardo Montolivo, Mattia De Sciglio, Ignazio Abate and Stephan El Shaarawy on the treatment table, but Allegri insists he is not unduly concerned.
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"I’m not worried at all about the situation," the 46-year-old continued. "This period that we’re going through, we’ve actually been a bit unlucky. We actually only have four players with what you can call muscular injuries. Then there are the traumatic injuries and here you’d need a crystal ball to see these things coming.
"The number of injuries we have reads like a wartime news bulletin but nevertheless, the work done by the medical staff and the physiotherapists has been outstanding."
Kaka suffered an abductor problem against Torino but agreed to waive his salary for the month he spends sidelined with the problem - something which Allegri feels is in keeping with the kind of player the Brazilian is.
"Kaka’s injury was getting worse and worse as the game wore on and the gesture that he gave yesterday regarding his salary shows the kind of guy he is and the respect he has for the club and the fans," he added.
Alessandro Matri is set to make his first Champions League appearance since joining the club from Juventus last season, and he is relishing the prospect of facing a side against whom he scored for the Bianconeri in 2012-13.
"It’ll be an emotional night. My Champions League debut in the San Siro - I can’t wait to get on the pitch and play this game," he said. "I played well against them last season so maybe that’s a good sign and let’s hope it happens again.
"I settled in quickly here. I knew a lot of the people already. If you’re playing for a big team, you can never have a place on the starting line-up guaranteed. But of course knowing that the coach believes in you, that the team believes in you, that everyone at the club is behind you, that really helps a player, especially when things are not going so well."