Police say Constable Farhad Hossain was critically injured when those attempting to take control of the centre opened fire.
The incident occurred at 9:30 am at the Chandanaish Government Primary School centre in the Sadar Upazila around 9:30 am.
A sub inspector named Shah Alam was also injured by brickbats hurled during the attack.
Farhad Hossain has been taken to Chattogram Medical College Hospital in critical condition.
Chattogram District Additional Superintendent of Police (South) Afrazul Haque Tutul told bdnews24.com that some men tried to take control of the centre around 9:30 am, leading to a clash with police.
“They were unable to take control of the centre. Voting has continued. But one of our constables has been shot.”
The OC said that the attackers had been identified as supporters of Awami League upazila chairman candidate AKM Nazim Uddin.
Nazim Uddin is competing against incumbent chairman Abdul Jabbar Chowdhury, who left the LDP and joined the Awami League, for the seat.
Voting is underway at 117 upazilas in 25 districts in the third phase of local elections on Sunday.