According the the union, Unite, London Underground faces the prospect of a complete shut down this Bank Holiday weekend as workers, who control the power to run the trains, prepare to strike.
Unite regional officer Hugh Roberts said:
"There is a real possibility that the underground could close down over the bank holiday weekend. Unite is working very hard to achieve a fair settlement for our members - and to avoid the industrial action. The ball is very much in the management's court."
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Kurt Cochran and his wife Melissa had been visiting London from the US to celebrate their 25th wedding anniversary and were due to fly home.
Rail union RMT have slammed the move saying the company MTR is set to make "a killing at the British taxpayers' expense".
The meeting was arranged in the wake of Wednesday's terrorist murder of four people in Westminster.