Milan captain Riccardo Montolivo was sent off late on in the contest while Sergio Pellisier also saw red in injury-time for the hosts, at the end of a drab encounter.
The Rossoneri are now without a win in their last six games and have recorded just three victories from their opening 12 league matches, which has heaped more pressure on Allegri.
"I am the coach of the Milan," he told Sky Sport Italia.
"So if the club thinks this team needs another coach then they'll let me know. I look at what happens on the field."
Robinho and Alessandro Matri both wasted good opportunities to give Milan victory and Allegri insisted he was satisfied with the display from his side.
"It was a good performance," he added.
"The lads played well, created a great deal and didn't allow many chances, which wasn't easy.
"We had Wednesday's game (the 3-1 Champions League defeat in Barcelona) in our legs and a situation that was difficult, but I could not have asked for more from my players.
"We needed to score, we didn't score, but if we had then the view of the overall performance would've changed.
"The lads created several scoring opportunities against a Chievo side that was clammed up into its own half, so I could not have expected more.
"When everyone gives their all, you cannot ask for more in terms of effort."