A long-running dispute over the role of rail conductors has dramatically worsened after workers were given a deadline for accepting changes or having their contracts terminated.

Southern Railway, which runs services through Watford, said it will press ahead with the controversial changes if workers do not sign up to the new post of on-board supervisor by midday on Thursday.

The company, which is facing a series of fresh strikes by the Rail, Maritime and Transport (RMT) union from next week, also invited passengers to tweet messages to the union telling them how they feel about the industrial action.

READ MORE: Southern faces Twitter backlash after urging commuters to 'strike back' at RMT.

Under the headline Let's Strike Back, the company took out newspaper adverts saying: "The RMT won't listen to us. But they may listen to you."

The move prompted an angry backlash from passengers, with most criticising the company rather than the union.

User Matt Woosie wrote: "I'd rather be delayed by a strike than your company's inability to hire enough staff."

Simon Cox posted: "You are kidding? I dislike unions but I dislike incompetent management more. You provide the service - get on with it."

Nik Fletcher said: "My connecting train home has been cancelled every day under revised timetable. And now you expect me to advocate for you?"

Southern said its Thursday ultimatum was a "final attempt" to resolve the long-running dispute, which has caused travel chaos for months.

The new offer included a lump sum payment of £2,000 to conductors to be paid once the dispute is settled and its plans are fully implemented, guarantees on conductors' jobs until 2021, above-inflation pay increases for the next two years and guaranteed levels of overtime.

In a letter to RMT leader Mick Cash, Southern said it will "regretfully proceed without the RMT's involvement" and serve notice letters to conductors, terminating their contracts and inviting them to sign up to the new on-board supervisor role from January if it does not agree to the offer by midday on Thursday.

Govia Thameslink Railway (GTR), which owns Southern, has also invited the RMT to meet for urgent talks.

READ: Three in ten Govia Thameslink trains 'fail to reach their destination'

Chief executive Charles Horton said: "Everyone is sick and tired of this pointless and unnecessary dispute and we now need to bring a swift end to these strikes, which have caused months of misery for hundreds of thousands of workers, children going to school, family days out and retired people.

"We have a responsibility to the travelling public and our staff and, after 10 months of dispute, these strikes are plaguing people's lives and enough is enough.

"The union and conductors have had an incredibly fair and comprehensive offer on the table for nearly two months, with job security for at least five years, pay increases and overtime guaranteed.

"Today we are going the extra mile and offering our conductors a lump sum cash payment to be paid just after Christmas ,when they are getting on with their new roles giving fantastic service to our customers.

"We've given the RMT and our striking conductors a fair, clear and unambiguous plan that we intend to implement if they won't do a deal.

"Independent experts have said that running trains with the driver closing the doors is safe and as a responsible operator we have always looked after customers with disabilities, and always will.

"No-one should be in any doubt that the deadlines are fixed and immutable and we will press ahead if there is no deal by Thursday lunchtime."

Mr Cash responded: "Jobs, safety and access on Southern rail services are not for sale for £2,000. This dispute has never been about money, it is about guaranteeing that there is a second, safety-critical member of staff on board the current Southern rail services.

"The union is ready for talks and has been throughout."

Mr Cash said there could be a deal based on an agreement the union recently reached with ScotRail that guarantees a second member of staff on all services.

"Despite the Southern spin, this dispute is not solely about the doors, it is about giving passengers that safety guarantee that goes with the second member of staff on the train.

"If that guarantee is good enough for Scotland, it should be good enough for Southern. RMT will be responding positively to the call for talks but we will not be caving in to ultimatums that put a £2,000 bounty on our members' heads and which try to bribe us into selling out on the core issues of rail safety and disabled access."

On the Twitter row, Southern said: "Our aim was to get the debate going and to let people know exactly what our very fair and comprehensive offer is and how we are trying to modernise the train service for our passengers, which the RMT leadership is trying to block."