Southern Railway is urging 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.
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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.
It comes as a Transport Select Committee report last week found that passengers had been badly let down.