Sources said that the TSSA union was going to suspend a walkout from 9pm tonight after reaching a deal during talks which ended at the conciliation service Acas today.
The move is expected to be confirmed later today.
The RMT union is also involved in the dispute, with its members due to take industrial action following a strike last week which crippled Tube services in the capital.
An Acas spokesman said: “Talks have adjourned at Acas this morning. London Underground will be writing to the trade unions with some proposals to end this dispute.
“We are hopeful of a positive response so that the proposed industrial action will be withdrawn.”
Workers were due to strike for 48 hours in protest at controversial plans to close all Tube ticket offices, with the loss of 950 jobs.
A 48-hour stoppage last week caused travel chaos in the capital.
The Federation of Small Businesses (FSB) estimated that last week’s action cost small firms in the capital £600 million in lost working hours, business and productivity.
The TSSA confirmed it had suspended its strike after reaching a last-minute deal at Acas with LU.
A spokesman said: “We have now agreed a process where all our serious concerns over safety and job losses will be seriously addressed through the normal channels.
“We are obviously pleased that we have agreed this process, which will allow us to suspend our strike immediately, and cancel it later when the agreement reaches us in a formal document.”
Comments: Our rules
We want our comments to be a lively and valuable part of our community - a place where readers can debate and engage with the most important local issues. The ability to comment on our stories is a privilege, not a right, however, and that privilege may be withdrawn if it is abused or misused.
Please report any comments that break our rules.
Read the rules here