According to the mayor's office, the boxes with the ballots from Baghdad's Rusafa district burned out.
The Iraqi Independent High Elections Commission rented the site from the Iraqi Trade Ministry. The security at the building has been stepped up, the outlet added.
On Wednesday, the Iraqi parliament adopted a bill stipulating that ballots from the vote should be recounted. Moreover, the legislation annulled the votes cast by internally displaced persons (IDPs) and canceled the results of the May 10 voting by Kurdish security forces personnel.
The electoral commission said it would appeal the parliament's decision to recount the ballots in court.
The May 12, the Iraqi parliamentary election were held in the country for the first time since Daesh defeat. The Sairun Alliance, backed by the prominent Shiite cleric Muqtada Sadr, won 54 parliamentary mandates, while the Conquest Alliance led by Hadi Amiri gained 47 seats, and incumbent Prime Minister Haider Abadi's Victory Alliance earned 42 seats in the 329-seat parliament.