The men were captured on CCTV cameras close to the Aleppo in Great Horton Road around the time it was damaged by fire.
The fire, which happened in the early hours of March 17, caused damage to the basement and is believed to have started inside.
A West Yorkshire Police spokesman today said: "Detectives at Bradford District CID have made extensive enquiries into the blaze and would ask anyone who may recognise one or either of these men to get in touch."
Contact police on 101, quoting crime reference 13180126715, or use the live chat facility on the West Yorkshire Police website.
Information can also be reported anonymously to the independent charity Crimestoppers on 0800 555 111.