Violence first broke out when two unidentified men on a motorcycle fired at a queue of voters, killing one person, the police official said.
Four people died near another polling booth when security forces opened fire to control unrest among voters, as people shoved and pushed duty police officers, said the official, declining to be named ahead of a final assessment of the incident.
A police spokesman declined to comment.
Dola Sen, spokeswoman for the state's ruling Trinamool Congress, told Reuters the death toll had gone up to eight as others who had been injured died later on Saturday.
Prime Minister Narendra Modi and his top ministers have campaigned hard to bring their Bharatiya Janata Party (BJP) to power in the big eastern state for the first time. They are facing a tough challenge from the Trinamool party, a regional group.
Previous rounds have been largely peaceful.
The BJP controls a dozen of India's 28 states, with alliance partners in several others. But it has never won power in Bengal, whose 90 million people make it the fourth most populous state. Wins in states dictate the position of a party in the upper house of parliament.