Millions of tube users will be left waiting until the last moment to find out whether there will be a full train service next week.
The RMT - the largest of the Tube unions - will not decide until Monday whether to go ahead with a 24 hour strike currently set to start on Tuesday evening.
The delay forced London Underground (LU) to start planning what sort of emergency service it may be able to run should the strike go ahead.
Travel advice and updates for passengers will be issued by LU “as soon as possible”.
Other major unions - Aslef, TSSA and Unite - have already confirmed they are suspending the strikes after receiving an improved offer in the long running dispute over the Night Tube, annual pay and working conditions.
RMT members include mostly station staff but also train drivers, engineers and signallers.
Mick Cash, the RMT leader, said a mass meeting of union representatives would take place on Monday followed by a meeting of the controlling executive committee which would then decide whether to go ahead with the walkout.
“To protect our position, the strike action due to start from Tuesday, 26 January, remains in place,” he warned.
The revised offer includes a four-year pay deal and plans for a 36-hour four day week.
Each of the unions warned that if future negotiations fail then there will be more strikes.
Night Tube was to have started on 12 September last year, with 24-hour trains on Fridays and Saturdays along the entire Jubilee and Victoria lines, but LU was unable to reach agreement on pay and working practices with the unions.
Steve Griffiths, LU’s chief operating officer, said it had put forward a “very fair offer.”