Should Celtic keep their Champions League dreams alive with victory over Milan, they would also plunge the Rossoneri even further into crisis.
Milan had cause to be grateful for some respite when landing in Glasgow on Monday evening. Some 400 furious supporters prevented the team bus from departing San Siro after the 1-1 draw with Genoa on Saturday. It took until midnight and for senior players, including Kaká, to address the irate crowd before calm was restored.
"Speaking to fans wasn't a huge burden for me," Kaká said. "It is not something I am used to but as one of the senior players I have to get used to responsibility."
The unrest is understandable. Milan sit 13th in Serie A, four points above the relegation zone and 20 adrift of the leaders, Juventus. Massimiliano Allegri's team have not won away from home all season or at all in their past seven matches.
"It is only normal that the fans are disappointed but they are not the only ones," Allegri said. "The staff and the players share that disappointment. We have to do better. We have three objectives: to qualify from the Champions League group stage, in Serie A we have to improve and we have the Coppa Italia. Any other problems, the players don't have to worry about. They just need to stay focused."
It seems apt that the infamous character of Mario Balotelli is caught in the midst of the turmoil. He and his team-mates have been branded "brats" by Milan's ultras, a theory hardly likely to subside amid reports Balotelli turned up late for training on Sunday.
On Monday morning the Milan vice-president, Adriano Galliani, who had pleaded with fans to ease off with their protests, insisted Balotelli would not be sold. For supporters of Manchester City, this will seem a familiar scenario.
"I think he'll enjoy it, you know," said the Celtic manager, Neil Lennon, of Balotelli's appearance in Glasgow. "I have seen him play at Old Trafford. City won 6-1 and he scored two and played brilliantly.
"He played brilliantly the other night despite all the negative headlines coming out of the game. He missed a penalty but his play in general was really good. He is so strong and his first touch was very good. I think he had 13 attempts on goal, didn't score but, with the fact he was making chances for himself or getting on the end of things, he looked a top, top player."
When reminded of Balotelli's negative alter ego, Lennon added: "I hope that Balotelli turns up tomorrow rather than the other one but you just never know."
In general, Lennon is wary of pointing to Milan's troubles as a reason for hope. "The Champions League is important for Milan in a lot of ways," he explained. "I just feel that with the players they have and the mentality of the club then if they can prevail in the group stages their form will pick up automatically.
"They have got a calibre of player and they like these nights. They like the Champions League and they have got the pedigree; they've got a backroom staff that have got four or five Champions League medals. So the whole structure of the club is geared towards the Champions League. It might not be a vintage Milan team but it's still an excellent side. They dominated the game against Genoa. For large passages of the game, Milan were excellent."
In the specific context of Group H, Celtic know that they must win to retain aspirations of reaching the last 16. Should that happen and Ajax fail to defeat Barcelona, Celtic will head to their final group match at Camp Nou with their fate in their own hands.
"It is win or bust for us," Lennon said. "I would like us to play really strongly at home and give the supporters something to really shout about. Whether that's enough on the night remains to be seen.
"I will be looking more to the performance than the result. If we win the game and don't play well I will still be obviously delighted. But I am looking for a really strong performance as well."
Lennon, who hinted he will hand his Dutch winger Derk Boerrigter a first Champions League start, will also have Kris Commons available. Commons' home was targeted by robbers as he slept during the early hours of Sunday, with two cars stolen from his driveway.
"It was not a nice thing for Kris and his family to suffer,"Lennon said. "But I have spoken to him and he is totally focused on the game. What happened can be traumatic but Kris, being the strong-minded individual that he is, has brushed it all off."