The 24-hour Tube had been due to start on 12 September, but sources say this target is likely to be missed.
This coincides with the news that train drivers will not be joining other London Underground employees in two fresh strikes later this month over the introduction of the night Tube.
Sources at the Aslef union said further talks will be held to resolve the long-running dispute over pay and rosters.
Members of three unions, the RMT, TSSA and Unite, are due to stage two 24-hour walkouts from the evenings of 25 and 27 August following two stoppages in the past month.
Len Duvall, leader of the London Assembly Labour group, criticised London Mayor Boris Johnson for his handling of the night Tube.
“By speculatively announcing a start date without any consultation with the people expected to run the service, Boris Johnson’s gung-ho approach has led to disputes, disruption and now delay,” he said.
“It’s been clear for a long time that the problems facing the night Tube would not be easily overcome, so in a sense a delay isn’t a major surprise. From the outset Boris Johnson has treated the night Tube as more of a publicity stunt despite it being a deeply complex project.”
London Underground said on Wednesday that if they met new union demands the additional cost would be £1.4bn by 2023/24 and could force a scaling back of modernisation plans or fare increases of up to 6.5 per cent.
Chief operating officer Steve Griffiths said: “Having previously argued that it was all about work-life balance, certain unions have now made a whole series of unaffordable demands for more pay, shorter working hours and the reversal of the modernisation of the Tube.”
The night Tube will lead to 24-hour weekend services on the Jubilee, Victoria and most of the Piccadilly, Central and Northern lines.