The dispute is over job cuts, pay freezes, working conditions and changes to contracts.
The voting will go on until February 22. If successful, a strike can be held by the 12,000 cabin crew of BA as early as in the first week of March.
BA has said it is committed to protecting its customers' travel plans by deploying temporary trained crew who can take over the roles of the normal crew.
The airline has begun a 21-day fast track programme to train non-cabin staff such as baggage handlers and technicians for replacing regular flight attendants, should the workers take industrial action to push their demands.
A Unite union official said this not only shows contempt for the crew, it also sends a wrong message to passengers who have paid to be cared for by a premier airline.
Unite said only 216 people had responded to last week's call by BA for volunteers to keep the airline running if the strike materializes.