Thousands of passengers who had their journeys disrupted because of the walkout are braced for further problems because of a four-day stoppage due to start on Saturday.
Tony Woodley, joint leader of Unite, said he had not received any positive response from the company to his plea for fresh talks and added that chief executive Willie Walsh's silence was "deafening."
BA said its contingency plans were continuing to work well around the world, maintaining that it had added extra flights at Heathrow and Gatwick because more staff than expected had turned up for work.
The two sides continued to clash over the impact of the strike, with BA accusing the union of publishing information which had "no basis in fact".
Among the claims made by Unite today were that some aircraft had left Heathrow with their blinds down to hide the fact that few, if any, passengers were on board.
It is understood Mr Woodley has contacted TUC general secretary Brendan Barber about restarting talks which collapsed last week after BA tabled an offer Unite complained was worse than a previous one.
Mr Woodley visited picket lines around Heathrow Airport today and addressed a rally, reaffirming that the union leadership remained ready for talks at any time before the next strikes.
"Willie Walsh's silence is deafening. Where's Willie? He has not been seen or heard from today while his business grinds to a halt," he said.
"Cabin crew have sent the company the most powerful message over the last three days that they will not be cowed or bullied into accepting industrial dictat. I am proud of their solidarity and resilience, and the support they have given to this dispute, which none of them wanted.
"I would like to hear BA's board justify spending millions on a floundering strike-breaking operation when they turned down an offer of more than £55 million in cost savings from their own cabin crew.
"We estimate that BA may have spent as much as £18 million on leasing airplanes over the last three days, and it is beyond dispute that most of its long-haul flights have been cancelled, and most of those which have taken off are half-empty or completely passenger-free.
"This is the economics of the madhouse which can only lead to suspicions that there is another, union-busting, agenda at work here.
"Trying to break Unite will break BA's bank account. That is why it is time to talk. Pick up the phone, Willie."
Striking staff were in a buoyant mood at the rally, many booing BA flights as they passed overhead and joining in chants aimed at Mr Walsh of: "Willie, Willie, Willie - out, out, out."
One held a placard mocking the airline's slogan, which read: "The world's favourite picket line".
Meanwhile, at Heathrow's Terminal Five, disgruntled passengers faced further frustrations.
German couple Carolin and Stefan Marquardt were stuck in the terminal building for seven hours waiting for a flight home to Stuttgart - having cut short their holiday to India by a night.
BA said: "We are operating our planned schedule of departures at Heathrow and Gatwick and we have been able to add flights into our schedule at both airports over the weekend as well as today.
"We continue to operate across the world to almost 120 different destinations and are continuing to add in extra cities, due to the numbers of crew reporting for work.
"Cabin crew are continuing to report as normal at Gatwick, and Heathrow levels remain above what we need to operate our published schedule.
"As a PLC, British Airways is legally obliged to ensure that it does not release information that is misleading or inaccurate.
"This information includes matters such as numbers of passengers we are able to carry, numbers of flights operated and numbers of crew reporting for work. Any suggestion in media reports that information we have issued is untrue implies that the airline's management is acting unlawfully.
"Unite and its cabin crew branch, Bassa, are under no such legal constraints. A great deal of the information they have put out over the last three days has no basis in fact. For example, Unite has no way of obtaining accurate figures as to how many customers are on our aircraft or how many crew are reporting for work."
Business group London First, whose members include many of London's internationally-based businesses, said the capital's reputation as a centre for global trade was being damaged by the strike.
Chief executive Baroness Valentine said: "Despite the best efforts of BA management and many staff to continue to put the interests of passengers first, the strike is reminiscent of a best-forgotten era.
"The message from BA's London-based business customers to both sides is: Get back to the negotiating table, and to union members: Get back to work in the meantime."
Comments