Wow this is the first time a question hit an area where I'm knowledgeable. Let's hope I meet the other criteria.
So without further ado to your question. Questions like these are generally hard to answer because we don't have many sources for individual opinions in Rome. We generally have a few writers or ancient historians, and circumstantial evidence like inscriptions and big results. This all skews our view of antiquity, especially for specific events. We don't have a voting record or a good grasp on what every senator was thinking. And it's all to easy to just lump the whole senate in one camp, but let's look at the events as far as we know them.
Caligula gets assassinated and leaves of course a power vacuum. The senate is summoned by one of the consuls (Quintus Pompeius Secundus). There a multiple possible claims to the position (Quintus Pompeius Secundus is one of them), but the senate fails to acclaim a new emperor (or at least what we now call an emperor, there isn't an imperial title in Rome, an emperor is emperor by accumulation of rights and other titles).
This gives us a first hint to the answer to your question. If the senators generally had thought of ending monarchical rule in favour of a return to a Republic, they wouldn't have tried to proclaim a new emperor. Generally everyone in the senate could at least hypothetically become emperor and we can assume that even if they framed the discussion as republican, many of them had secret ambitions.
For a short time there is no emperor and the consuls take in essence republican responsibilities and for example give the urban cohort their watch word. The urban cohorts however express that they want a new emperor and the search for a new emperor didn't stop.
Claudius, Caligula's uncle, takes possession of the cesarean household and with it all of its abundant resources that aren't that strictly separated from the resources of the "empire". Invoking posessio, in essence not to far from modern squatters rights, he becomes head of the cesarean household and proclaimed emperor by the praetorian guards. The senate reconvenes and accepts that the praetorians support Claudius' bid for emperor and confirm his position.
If they had wanted to return to the Republic, they could've refused to bestow their approval. This however may have been a bad choice considering the praetorian guard (4000-6000 men) in town.
This leaves a "probably no" as an answer which is further confirmed by the fact that the senate through centuries at least consented through action and inaction to a monarchical system. This doesn't mean that the senators liked every individual monarch, but they arranged themselves with the monarchical system.
Sources