The names of Dutchman Frank Rijkaard, former Barcelona coach Pep Guardiola and assistant coach Mauro Tassotti have been thrown into the rumour mill by the Italian media following Wednesday's Champions League defeat at Malaga.
Whether any of them could do better than Allegri with such limited resources is highly debatable, especially with the pressure to get instant results.
Allegri, who led Milan to the 2010/11 title in his first season there, has been forced to rebuild his side after the club sold off Zlatan Ibrahimovic, Thiago Silva and Antonio Cassano during the close season in a bid to balance the books.
With Brazilian forwards Robinho and Alexandre Pato struggling with injuries, Allegri has turned to Stephan El Shaaraway and, although the teenager is a bright prospect, it seems premature to ask him to lead the attack.
A loss of form by right-back Ignazio Abate and midfielder Kevin-Prince Boateng, left out of Wednesday's team for technical reasons, has not helped the cause either.
Milan have slumped to 15th in Serie A with seven points from eight games and both Allegri and chief executive Adriano Galliani have repeatedly had to fend off questions about the coach's future.
"We all know we are in a difficult moment, I am the first to realise that, but I also say this team will recover due to the efforts they are putting in," said Allegri after Wednesday's 1-0 defeat in Spain.
"How do we get out of this? We beat Genoa on Saturday. It's the only course of action."
Genoa, who have two points more than Milan, will be playing their first game under new coach Luigi Del Neri after Luigi De Canio was fired on Monday following their 4-2 home defeat to AS Roma.
Del Neri, who has coached more than 15 Italian teams including Juventus, Roma and Sampdoria, became the sixth coach to be appointed by volatile club president Enrico Preziosi in two years.
Leaders and title-holders Juventus have also had an uncomfortable week. Their Champions League campaign stuttered with a 1-1 draw at Danish champions Nordsjaelland and striker Niklas Bendtner spoke of his frustration at being repeatedly left out of the team.
"I have the quality to be a part of this great club," he said in a Danish television interview widely reproduced by Italian media.
"I don't see why I shouldn't play and I hope to soon be a first-team starter. I can play in this team and I've shown that. I'm not at 100 percent, but I'm convinced that I can do well."
Juventus, who have 22 points from eight games, are away to Catania's heavily Argentine-influenced side in Sunday's early kick-off.
The Sicilians, who have made a respectable start to the season with 11 points, will attempt to end Juve's 47-match unbeaten run in Serie A which stretches back to the end of the 2010/11 season.
Napoli, second with 19 points, host Chievo on Sunday while Lazio, a further point behind in third, visit Fiorentina.
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