One of the unions representing London Underground workers has already rejected an offer aimed at averting a fresh strike over the new all-night Tube.
Unite, whose members include engineers and power staff, said the proposed deal did not go far enough, especially on the flexibility of working the new night Tube.
The move increases the likelihood of the three main rail unions turning down the offer.
Thousands of workers are due to walk out for 24 hours from Wednesday evening, which would shut down the service for the second time in a month.
The unions are unhappy at the pay and conditions over the night Tube, which is set to be launched on September 12.
LU made a revised offer last week which will be discussed by union leaders before the two sides resume talks at the conciliation service Acas.
The new offer has not been increased in value but LU said it had been "reshaped" in response to concerns about the work-life balance of staff who will work on the new service.
Meanwhile, Transport for London has awarded a £760 million contract for modernising the signalling and train control system on four Tube lines to global transportation systems provider Thales.
TfL said the next major phase of the Underground's modernisation will bring faster, more frequent and reliable journeys to millions of passengers who use the Circle, District, Metropolitan and Hammersmith & City lines.
Up to 1,100 jobs and 60 apprenticeships will be sustained or created in London and across the UK.
The improvements will boost capacity by an average of a third on the four lines. Work is expected to begin later this year and the main benefits will be delivered by 2022, when the frequency of trains running during peak periods will increase to 32 trains an hour in central London.
Nick Brown, managing director of London Underground, said: "Having successfully modernised three of the most heavily used lines on our network, we are ready to begin work to bring the next four lines in to the modern era.
"This will transform the journeys of millions of our customers, significantly increasing service reliability and frequency."