Unite union joint leader Tony Woodley has urged British Airways' Chief Executive Willie Walsh to change his "petty, vindictive" attitude to secure an agreement and avert 15 days of strikes.
The leaders are meeting in Central London in last-ditch talks to prevent the industrial action planned by Unite over pay and working conditions for BA cabin crew.
The talks are being mediated by Acas, who have described this weekend as a "window of opportunity". The body says if no resolution is made after this weekend, the whole negotiation process will be problematic going forward.
BA boss Mr Walsh says the union members are "out of touch with reality", but claims he has made concessions which mean that cabin crew who have taken part in industrial action will not have their travel concessions taken away from them.
If the strikes, which are due to begin on Monday, go ahead - it will cost BA an estimated £100 million. The airline needs to avoid this after posting record losses of £531 million on Friday.