State exit polls have suggested the changes will be backed by over two thirds of voters, who have been encouraged to vote with prize draws offering flats and an ad campaign highlighting other amendments designed to appeal to the populace.
One amendment guarantees inflation-linked pensions; another says marriage is a union between a man and a woman.
One-off payments of 10,000 roubles ($141) were distributed to those with children at Putin's order as people headed to polling stations on the last day of the vote, held over seven days to try to limit the spread of the new coronavirus.
Putin, a 67-year-old former KGB officer who has ruled Russia for more than two decades as president or prime minister, made no mention of how the changes could affect his own career in an eve-of-vote speech on Tuesday (June 30).
The amendments would allow him to run for another two six-year, back-to-back stints if he wishes after his current term expires in 2024.
Putin has said he has yet to decide on his future.