British Airways could face even more strike disruption this summer unless it resolves a dispute with cabin crew over travel perks, the co-leader of Britain's biggest union Unite said on Monday.

Cabin crew began a second five-day strike on Sunday, ahead of a week of school holidays, after talks between Unite and the airline failed to reach an agreement last week.

Both sides in the dispute have claimed a deal over pay and cuts to staffing levels and travel could be close but blame each other for the lack of further progress. Unite have offered to suspend the walkouts if the airline restores travel perks stripped from striking crew.

A third wave of strikes, which stem from a long-running dispute over the airline's cost-cutting drive, are due on June 5, days before the start of the World Cup soccer tournament in South Africa.

Tony Woodley, the joint general secretary of Unite, told delegates at a union conference on Monday that a new ballot for continued industrial action could be only a week or so away.

There is only one thing to do with bullies -- that is stand up to them until they learn some manners, Woodley said, according to extracts from his speech issued by the union.

Woodley addressed BA chief executive Willie Walsh directly in his speech.

We all know there is a deal to be done at British Airways, one that recognizes the real commercial needs and problems of your company as well as our members' legitimate interests. Unite is ready to do that deal, he said.

But we are not, and never will be prepared to see our members and our union humiliated, victimized and reduced to ruins.

The stoppages have happened at a difficult time for the airline and come on top of seven days of walkouts in March which cost BA 43 million pounds ($62.75 million).

Earlier this month BA reported a second straight year of record losses as it battles a global economic downturn and industry-wide recession as well as disruption caused by volcanic ash drifting over Europe from Iceland.

(Editing by Greg Mahlich)

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