Rail strikes are already planned on October 1 and 5, threatening fresh travel chaos for passengers.
RMT general secretary Mick Lynch said it was encouraging that the new Transport Secretary Anne-Marie Trevelyan had met the union.
“We welcome this more positive approach from the government to engage with us as a first step to finding a suitable settlement.
“However, as no new offer has been tabled, our members have no choice but to continue this strike action.
“We will continue to negotiate in good faith, but the employers and government need to understand our industrial campaign will continue for as long as it takes,” he said.
The affected train companies are Chiltern Railways, Cross Country Trains, Greater Anglia, LNER, East Midlands Railway, c2c, Great Western Railway, Hull Trains, Northern Trains, South Eastern, South Western Railway, Transpennine Express, Avanti West Coast, West Midlands Trains.
A Network Rail spokesperson said: “We have received notification of another strike by the RMT on Saturday October 8, just one week after another walkout it has co-ordinated with train drivers at Aslef.
“This latest strike will again mean very significant disruption for passengers, and we’ll be asking people to only travel if absolutely necessary due to the reduced service that will be in place.
“Full timetables for all upcoming strike days will be published in due course.”
The RMT had already previously announced a walkout for October 1, potentially disrupting travel to the governing Conservative Party's conference in Birmingham in central England which begins on October 2.
The Transport Salaried Staffs Association has already announced it will ballot hundreds of its members at Govia Thameslink Railway (GTR) over possible industrial action, including strikes. The union represents workers including platform and ticket office staff, train crew, engineers and management.
It follows a summer of prolonged industrial action by the RMT as part of a long-running dispute over pay and conditions with Network Rail.
The RMT staged three days of industrial action in June, with further strikes taking place throughout July and August.
Previous strike action due to be undertaken by the union on September 15 and 17 was cancelled following the death of the Queen.
Speaking in the US on Wednesday, Prime Minister Liz Truss called for an edn to the strikes.
She said: “We are committed to bringing in legislation for minimum service levels on rail as soon as possible.
“My message is: I want this country to be successful. And that means people being able to get to work. People being able to get on with their business, people being able to move projects forward.
“So I would encourage rail workers to get back to work. There’s no doubt we’re facing tough times as a country. I want to take a constructive approach with the unions, but I would tell them to get back to work.”
The Department for Transport warned that the unions were taking the railway system back to the “dark ages”.
A spokesman said: “Thousands of people once again will have their day-to-day lives disrupted and be unable to attend work, school or doctor’s appointments.
“Our railway is in desperate need of modernisation but all more strikes will do is take it back to the dark ages and push passengers further away.
“We urge union bosses to reconsider this divisive action and instead work with their employers, not against them, to agree a new way forward.”