The train company has offered its staff a £2,000 lump sum to agree to new contacts, and has asked the RMT Union to consult their members about the deal.
Staff are due to walk out from midnight tonight for the latest series of strikes.
But Southern has said that despite the dispute more conductors are turning up for work during the 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.’
In a letter to RMT leader Mick Cash, GTR human resources director Andy Bindon said: ‘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.’
The union had rejected a £2,000 bonus for conductors during previous talks.
There are 11 further strike dates planned between tomorrow and Christmas, with the first series ending on Thursday.