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Hopes of a settlement to a long-running rail dispute rose on Thursday after a meeting between Mick Lynch, general secretary of the RMT rail union, and transport secretary Mark Harper.

The RMT, the UK’s biggest rail union, on Tuesday announced an overtime ban for its members at Network Rail and 14 train operators for the festive period, as well as two 48-hour work stoppages in December and a further two such walkouts in January.

Lynch declined to call off those strikes after the meeting.

“It was a positive meeting, in the sense that we got rid of the bellicose nonsense we used to get from Grant Shapps,” Lynch said, in reference to the former transport secretary who refused to meet him.

In a meeting that he described as “constructive”, Harper said he and Lynch held an “open and honest conversation” about the “serious challenges facing the railways”.

“We have common ground — we both want the dispute to end and we both want a thriving railway which delivers for passengers and workers alike,” Harper said.

To achieve that, however, all parties needed to work together, Harper added.

“There is a deal to be done, and I believe we will get there,” he said. “I want to facilitate the RMT and the employers to reach an agreement and end the dispute for the benefit of the travelling public.”

Harper on Thursday morning told the House of Commons he would meet representatives of rail unions “in the coming days”.