Manuel Cortes, general secretary of the Transport Salaried Staffs' Association (TSSA), said: "The clock is ticking and we need an agreement by the end of this week or sadly, further strikes will follow."
A 24-hour walkout held by members of the TSSA and the Rail, Maritime and Transport (RMT) union in a dispute over staffing levels at stations caused travel chaos earlier this month when the majority of Zone 1 stations were shut.
Talks continue today at conciliation service Acas in a bid to avert further walkouts.
Cortes said: "Our negotiating team remains committed to reaching a resolution which ends this long-standing dispute. We want to ensure that Boris's last act of vandalism on Londoners is undone so that our Tube is once again safe for both passengers and staff."
The RMT previously warned industrial action will be escalated from Monday 6 February, if a resolution isn't secured.
Former mayor Boris Johnson agreed to the closure of the ticket offices, which the unions argue has left the Tube understaffed.
His successor Sadiq Khan commissioned an independent review into the closure, which found that offices shouldn't be reopened but they had been closed prematurely and said measures needed to be taken to make staff more visible and improve ticket machines.
While Transport for London (TfL) said it had always agreed to review staffing levels and started to recruit another 150 members of staff, the unions said this wasn't enough.
Steve Griffiths, London Underground's chief operating officer, said last week he hoped the talks at Acas would "make progress towards resolving this dispute".