Partial results, announced five hours before polls closed, indicate the former KGB officer who has ruled Russia for more than two decades as president or prime minister will win the right to run for two more terms. That means he could remain president for 16 more years.
The Central Election Commission said just over 70% of votes counted across the world's largest country had supported changing the constitution. Almost 29% had voted no of the 2.68% of ballots counted. Fuller results are due at 1900 GMT.
Russians have been encouraged to vote with prize draws offering flats and an ad campaign highlighting other constitutional amendments in the same reform bundle, such as the pensions protection and a de facto ban on same-sex marriages.
One-off payments of 10,000 roubles ($141) were transferred to those with children at Putin's order as people headed to polling stations on Wednesday, the last day of the vote, held over seven days to try to limit the spread of the coronavirus.
"I voted for the amendments to the constitution," Moscow resident Mikhail Volkov said. "We need radical changes and I'm for them."
Others voted for the changes with less enthusiasm.
"I didn't read about the amendments if I'm honest," another voter, Lyudmila, said. "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 had reached nearly 60% by midday, election officials said. The required turnout is 50% and the amendments will pass if they are backed by a simple majority of voters.
Putin, 67, made no mention of how the changes could affect his own career in an eve-of-vote speech on Tuesday. They would allow him to run for another two six-year, consecutive stints after his current term expires in 2024.
Putin has said he has yet to decide on his future. Critics say they are sure he will run again, but some analysts say he may want to keep his options open to avoid becoming a lame duck.
At 60%, according to the Levada pollster, his approval rating remains high but well down on its peak of nearly 90%.
With Russia reporting thousands of new COVID-19 cases each day, opponents have been unable to stage protests but have mocked the vote online, sharing photographs of polling stations in apartment stairwells, courtyards and the boot of a car.
SMALL PROTESTS
A small group of activists staged a symbolic protest on Red Square on Wednesday using their prostrate bodies to form the date - 2036 - before being swiftly detained by police, TV Rain reported.
The "No! Campaign," called on supporters to vote against the changes and then discuss the result on Moscow's Pushkinskaya Square while respecting social distancing rules.
"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," Andrei Pivovarov, an activist, said in an online video.
Mass gatherings are banned in Moscow due to the virus and Pivovarov's suggestion of a public discussion was carefully worded. He avoided calling on people to gather, saying only that he wanted to discuss the vote's outcome "with other free people" and would be in the square on Wednesday evening.
More than a dozen police vehicles were parked in the square beforehand, where a lone masked protester stood with a sign saying "I/We are the constitution".
The opposition Communist Party, which has advised supporters to vote "no", has complained of irregularities.
Gennady Zyuganov, its leader, said that Putin and voters needed to weigh the consequences of sticking to the Russian leader's policies, which he said had failed the economy.
"Putin needs to choose," said Zyuganov. "For him the moment of truth is coming."
Putin has said he wants a clean vote, something election officials have pledged to deliver.
Golos, a non-governmental organisation that monitors elections, has said it will not be able to confirm the outcome of the vote as legitimate.