Southern Railway has urged union leaders to put its offer to a referendum to end the dispute over the role of conductors, saying a £2,000 lump sum is back on the table if they do.
The move came ahead of a fresh round of strikes, with members of the Rail, Maritime and Transport (RMT) union due to walk out for three days from midnight.
Southern said increasing numbers of staff were turning up for work during strikes.
Charles Horton, chief executive of Southern's owner, Govia Thameslink Railway, said: "The RMT needs to reflect on the hardship, distress and frustration being experienced by our customers and employees. Everyone wants to see an end to this dispute, but it is clear that, as things stand, there is currently little prospect of a negotiated settlement between us, however much we've tried.
"There have been significant developments since the union tabled the original ballot to conductors over six months ago. In order to move things forward, I am asking to union to let conductors - the people at the heart of this matter - determine whether this dispute comes to an end or continues.
"Nearly a third of conductors are now coming in to work on strike days and we expect that to increase. There is a full and fair offer on the table and it's time to stop the strikes and move forward."
GTR human resources director Andy Bindon said in a letter to RMT leader Mick Cash: "Given the RMT's democratic credentials and pride in being a member-led union, the fundamental changes made since March, both overall and in the RMT's position and in the absence of our being able to secure a negotiated settlement, you cannot have any reasonable grounds for refusing your members a vote.
"Needless to say, if, as I hope, you agree, then all planned industrial action should be suspended pending the outcome of the referendum."
Mr Bindon said the dispute started last December, leading to a year of "uncertainty and worry" for conductors and a strain on industrial relations.
The dispute, which has seen 12 days of strikes since April, has brought "disruption and misery" to passengers, he said, adding that the offer aimed at breaking the deadlock contained "unprecedented" guarantees and assurances.
Mr Bindon said the RMT's position has changed following the union's decision to advise conductors to accept new contracts moving them to new posts of on-board supervisors.
The union has rejected the offer of a £2,000 bonus for conductors during previous talks.
Mr Bindon said: "The fact is, that we are now implementing the changes we need to make to our train services and on-board roles in order to modernise our service and improve our customers' experience. These changes will happen, as is recognised by your advice to members to sign their contracts.
"Our employees and customers are therefore struggling to understand the statements made by some of your officials that strikes will continue into the new year, once the changes are implemented. No-one is losing a job, no-one is losing pay, guarantees have been given on numbers employed; it is a safe method of operation.
"You will no doubt be aware that nearly a third of conductors overall came to work during the most recent strike action, and in some depots around two-thirds worked normally. The numbers of conductors working normally is approaching a level that all commentators recognise makes long-term strike action unsustainable."