The incident happened in Shahbazpur village in the Domkal police station area around 10.30 pm on Wednesday, they said.
The CPI(M) alleged that its member Kader Mondal (42) was killed, and Asim Al Mamoon (43) and Lal Chand Mondal (42) were injured after TMC candidate Jafikul Islam run his car over them.
Islam, the TMC''s Domkal candidate, rubbished the allegation, claiming that he was far away from the spot when the incident happened.
Police said an investigation has been started to identify those behind the incident.
The Chief Election Officer (CEO) sought a report from the district authorities over the incident.
"We have asked the District Election Officer to send a report about the incident. Those behind this will be nabbed soon," an official at the CEO''s office said.
The injured are undergoing treatment at the Murshidabad Medical College and Hospital.
Tension was palpable in the village following the incident.
A huge contingent of central forces were deployed in the area to prevent any flare-up as polling started at 7 am, an officer said.
Elections were underway in Domkal in the eighth phase.