The Rossoneri have won just one of their first four games of the season and on Sunday went down 2-1 to Udinese, increasing the pressure on Allegri.
However, chief executive Adriano Galliani insists both he and owner Silvio Berlusconi have faith in Allegri.
In quotes reported in La Gazzetta dello Sport, Galliani said: "Allegri has my confidence and that of president Berlusconi.
"Milan played better, absolutely, I saw a team improvement.
"Now we look for our first victory on Wednesday although this year the San Siro seems bewitched for us and for Inter."
Udinese went ahead through a Mathias Ranegie goal just before half-time before Stephan El Shaarawy levelled.
But as Cristian Zapata became the first of two Milan players to be dismissed, Kevin Prince-Boateng later walking for two yellow cards, Antonio Di Natale converted from the penalty spot to earn the Zebrette their first three-point haul of the new campaign and cast Allegri's future into serious doubt.
Galliani added: "El Shaarawy's goal was very nice but then we conceded a penalty.
"Let's go ahead with this manager, 162 points over two seasons is a lot.
"We are only at the fourth day.
"The team suffered a defeat playing a good game - it's the sad consolation.
"The team played good football, but must be serene. We have all the qualities to recover."
Despite Allegri having received the backing of the club's hierarchy, there was bad news for the coach later on Monday when it was announced he had been suspended for one game for insulting the referee following the game.
Boateng was also awarded a one-match ban following his red card.