His side have only won one of four matches and have failed to score at the San Siro, where they were booed off after a goalless Champions League draw with Anderlecht.
Italian media reported a furious exchange between Allegri and former striker Inzaghi, who is in charge of the youth team and has been tipped to replace the former Cagliari boss in the event of his sacking.
But Allegri moved to deny the seriousness of the row – even going on to praise Inzaghi as a future top coach.
"Nothing happened, between me and Pippo there is no problem,” Allegri told Milan TV.
“There was only an exchange of opinions without any offence, unlike how it was written.
"I hope to stay at Milan a long time. Filippo has all the characteristics of someone who can have a good career as a manager."
Club vice president Adriano Galliano said the matter was closed for him.
"I have spoken to both Allegri and Inzaghi," Galliani told Gazzetta delloSport. "They have both played down the incident.
"As far as I am concerned, the incident never took place."
Allegri won the Serie A title in his first season as Milan boss, coming second to an unbeaten Juventus side last time out.
However, budget cuts by the club’s owners have led to a complete restructuring of the first-team squad, leaving Allegri at the mercy of notoriously impatient Italian fans.