The country’s Central Election Commission said 74 percent of the 30 percent of ballots which have so far been counted supported the proposed changes, with just under a quarter voting no. Victory for Mr Putin would mean the former KGB officer, who has led the country in one guise or another for more than two decades would be permitted to run for two more terms as President, meaning the 67-year-old could be able to carry on as President for 16 more years.
By comparison, Stalin served as General Secretary of the Communist Party of the Soviet Union – and therefore leader – from 1922 until his death in 1952.
Mr Putin has said he has yet to decide on his future.
However, critics say they are sure he will run again, although some analysts believe he may want to keep his options open to avoid becoming a lame duck.
At 60 percent, according to the Levada pollster, his approval rating remains high but significantly down on his peak of almost 90 percent.
Andrei Pivovarov, an activist with the No campaign, said in an online video: “We need to remind the authorities that we exist and that there are tens of millions of us who do not want Putin to rule until 2036.”
A small group of activists staged a symbolic protest in Red Square today, with one holding a sign saying “I/We are the constitution.”
READ MORE: Manchester coronavirus PANIC: Fears as infection rate six times higher
Moscow resident Mikhail Volkov said: “I voted for the amendments to the constitution.
“We need radical changes and I’m for them.”
Others voted with less enthusiasm, with one, Lyudmila, saying: “I didn’t read about the amendments if I’m honest.
“What’s the point of voting if they’ve already decided for you.
“It’s like that in our country – read something and vote. I voted.”
Turnout was roughly 65 percent, election officials said. The required turnout is 50 percent and the amendments will pass if they are backed by a simple majority of voters.
The opposition Communist Party, has urged supporters to vote “no”, alleging voting irregularities.
Leader Gennady Zyuganov said Mr Putin and voters each needed to weigh the consequences of sticking to the Russian leader’s policies, which he said had failed the economy.
He added: “Putin needs to choose. For him, the moment of truth is coming.”
Mr Putin had said he wanted a clean vote, something election officials had pledged to deliver.
However, Golos, a non-governmental organisation which monitors elections, has said it will not be able to confirm the outcome of the vote as legitimate.