If approved, the constitution would limit the president to two five-year terms, the BBC reported.
The document would not be retroactive, however. President Robert Mugabe, who has held office since the country gained independence in 1980, can run for re-election in July.
Mugabe's Zanu-PF party has shared power with the Movement for Democratic Change (MDC) in a coalition government since the 2008 election, which was marred by violence.
Some violence was reported in the run-up to this weekend's voting -- including an attack by several youths on a member of MDC as he was putting up posters, and a firebombing Tuesday at the home of a Zanu-PF official.
The MDC has focused its campaign on clauses in the proposed constitution that would guarantee protection from violence and free and fair elections.
Mugabe "wants the people to vote in peace," said Zanu-PF spokesman Rugare Gumbo.
The draft charter outlines citizens' rights and sets up checks and balances for people in power.
While Western and U.S. observers have been banned, the Zimbabwe Election Support Network has deployed about 600 poll watchers. About 2,000 local and foreign observers have been accredited.