Southern is close to agreeing a final deal with its train drivers in a dispute over pay and working arrangements.
ASLEF held several strikes in December and January leading to the complete shutdown of the Southern network run by Govia Thameslink Railway.
The union joined the RMT in a dispute over driver-only operation, which makes drivers responsible for opening and closing train doors.
A spokesman for ASLEF said: “Further discussions between ASLEF and GTR have taken place today [Thursday] and significant progress has been made.
“We are now in the process of finalising a proposal for agreement.”
Previous deals negotiated by the union’s leadership have been rejected after a ballot of its members.
Meanwhile the RMT, which has raised concerns over safety and access, is due to hold fresh strikes on Tuesday October 3 and Thursday October 5.
THe union is also balloting staff working for Churchill Service Solutions, which holds the contract to clean GTR’s trains.