There was never a serious push to return to a Republic. There are a couple factors that play into this:

• The Roman "Republic" was never really a republic as you might think of it: the Roman Senate had little formal power compared to a modern legislative body: they didn't even pass laws (there's a great post on this here). Much of the power was instead vested in a handful of magistrate offices culminating with the two consuls, who held very large amounts of power. These guys (and most of the other offices) were elected posts, but what's important is that power was relatively concentrated even from the start of the Republic.
• Even in the late Republic, power was starting to centralize around individuals rather than the state, and generals would frequently raise their own armies and fight their own wars. This trend was continued and amplified under Augustus, who further centralized power, to the point where he claimed about half of the Roman provinces to report directly to him.
• Augustus managed to reign for over four decades, which is pretty astonishing when you think about it. What this means is that most people had no conception of living in a pre-Imperial world, to them the emperor was a normal facet of the government. When Augustus died, it seemed almost obvious that there would be a new emperor.

I'd say the closest they got to returning to a Republic was immediately after Nero's death. Nero was the last emperor who was directly related to Augustus, which meant after he committed suicide Rome ended up in the unprecedented situation where there was no heir to take the throne. This led to a giant power vaccum--but the Senate never seriously attempted to fill the void. The Republic had been dead for 100 years and anyone who lived under the Republic was dead as well. instead we got the year of four emperors.

And even if they watned to, the Senate arguably wouldn't be able restore the republic anyways, because they didn't have a professional army backing them like all the claimaints to the Imperial throne.

Augustus is said to have twice thought of restoring the republic

"He twice thought of restoring the republic; first immediately after the overthrow of Antony, remembering that his rival had often made the charge that it was his fault that it was not restored; and again in the weariness of a lingering illness, when he went so far as to summon the magistrates and the senate to his house, and submit an account of the general condition of the empire.Reflecting, however, that as he himself would not be free from danger if he should retire, so too it would be hazardous to trust the State to the control of more than one, he continued to keep it in his hands; and it is not easy to say whether his intentions or their results were the better." - Suetonius, The Lives of the Caesars, The Life of Augustus.

It should be added that in the late republic the number of people voting in the comitia dropped to a ridiculous level, to the point it almost blocked the voting system. In addition the army had a lot to do with the designation of the new emperor and they kept supporting their favourite, the senate had not such a possibility, except to choose their own emperor (like Nerva after the assassination of Diocletian) and hope that no general would try to take power.