A man casts his ballot for a referendum at a polling station in Mariupol on September 27, 2022. AFP
The votes, which could result in Russia annexing those areas, have been branded an escalation of Moscow's campaign and called a "sham" by Kyiv and its Western allies.
Here's what Russian state media say will happen next, based on Russia's annexation of Ukraine's Crimean Peninsula in 2014:
Just as the annexation of Crimea has never been recognised by the international community, Kyiv and its allies have vowed not to accept the results.
The Kremlin, however, is keen to give the polls a veneer of respectability by following a superficially legitimate voting process.
There will be a period of ballot counting, after which "electoral commissions" set up by the Kremlin in each of the four regions will announce "provisional results".
These are expected on Tuesday evening or later this week.
Russian parliament's role
If the regions vote to be annexed by Russia -- which is a foregone conclusion -- the Russian parliament, the State Duma, will approve a treaty formally incorporating the four regions into Russian territory.
Over the weekend, Russian state news agencies TASS and Ria Novosti quoted parliament sources as saying an annexation bill could be proposed Tuesday evening and approved by the Duma on Wednesday.
The speaker of parliament, Vyatcheslav Volodin, said last week he would "support" the integration into Russia of the regions -- Lugansk and Donetsk in the east and Zaporizhzhia and Kherson in the south.
The bill will then be approved by the upper house of parliament, the Federation Council. This formality could be completed on Wednesday or Thursday, Russian news agencies say.
Declaration by Putin
Russian President Vladimir Putin is expected on Friday to formally declare the Ukrainian regions have become part of Russia, according to Russian news agencies.
This could take the form of an address at the Kremlin to the members of one or both houses of parliament.
TASS said senators had been told to take three Covid-19 tests in preparation for an "important event" on Friday -- a prerequisite for everyone who encounters the president at the Kremlin.
There is a possibility, however, that Putin will make an announcement before Friday.
In 2014, he signed a treaty incorporating Crimea into Russia just two days after the Kremlin held a referendum on annexation there.
The signing, at a special ceremony in the Kremlin, took place before the draft treaty had even been submitted to parliament.