Club president Massimo Moratti has in no way hinted he is pondering such a drastic move but did signal his unhappiness following last weekend's derby defeat by leaders AC Milan, who now have a six-point advantage over fifth-placed Inter.
"It is not a terrible situation, even if of course it isn't great because the derby is the derby and I think it should be played with a different mentality," Moratti told reporters.
"Everyone needs to get their head down, from the manager to everyone else, that way the team can start working. There are big margins for improvement."
Benitez, already with much to prove following a dreadful final campaign at Liverpool last season, has pointed to the array of injuries to key players such as Diego Milito, Walter Samuel and Maicon but the team have appeared to lack hunger this term.
Having won a stunning treble last season under their beloved Jose Mourinho, Inter were never likely to reach the same heights after the Portuguese moved to Real Madrid.
But Moratti, implored by Benitez to buy in the next transfer window in January, had assumed the same squad with no additions could still flourish.
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Instead, apart from the in-form Samuel Eto'o they lack the driving runs into the box or confident passing which characterised last season with even ultra-reliable captain Javier Zanetti giving the ball away in the derby.
Mid-table Chievo have 16 points from 12 games and do not offer an easy respite for Benitez but he has to get his injury-hit side into gear now to avoid more heat from Moratti.
"We all have to give more," the Spaniard said.
While rivals Milan bask in their new-found position as scudetto favourites after five years of Inter success, playmaker Ronaldinho is not quite as buoyant.
The Brazilian was left on the bench for the derby, despite Filippo Inzaghi and Alexandre Pato being injured, with Milan coach Massimiliano Allegri quickly impressing pundits by his handling of big players despite his lack of experience.
"I won't give up. I'm happy at Milan," Ronaldinho said, keen to return to the side and shine in Saturday's home game with Fiorentina.
Second-placed Lazio, a point behind Milan, visit quickly improving Parma on Sunday.
