"I don’t believe the British people are safe with this government running the NHS," Dr Andrew Meyerson said.
If it has a strategy, No.10 is holding its cards close to its chest.
Union bosses say 97% of workers were in favour of industrial action, where unpaid overtime and lengthy shifts on constant "high alert" are the norm.
The latest move comes after two days of strike action, on September 9 and 10, over a dispute about pay.
Pilots have walked out for two days, in the first strike to ever hit the airline. More than 1,700 flights have been cancelled.
BA has offered a pay rise of 11.5% over three years but Balpa says its members wanted a bigger share of the company’s profits.
Pilots will walk out on Monday and Tuesday due to an ongoing dispute over pay.
The airline said it was ‘likely’ anybody booked to fly on September 9, 10 and 27 will not be able to travel.
Thousands of people received emails overnight.
The two sides have been meeting throughout the weekend under the chairmanship of the conciliation service Acas.