The transport secretary, Grant Shapps, has written to the head of the rail union representing tens of thousands of striking rail workers to demand that he put a pay offer to a members’ vote. Shapps wrote to Mick Lynch, the general secretary of the Rail, Maritime and Transport workers’ union (RMT), after Lynch warned earlier on Saturday that further rail strikes were “very likely” after the latest wave of industrial action, as transport services were disrupted by the latest strikes over pay and conditions. Football fans and festivalgoers were among those affected by strikes on Saturday by tens of thousands of rail workers which were estimated to have left just 20% of the networking running. Lynch said that public support for action by his members was “entrenching” as he rejected suggestions they would agree to an offer already on the table if it was put to a vote. “I’ve got to say that it’s very likely given the gap between us at this time,” he told BBC Breakfast on Saturday, when asked if the union could take more action at a later date on top of what was already scheduled. In a letter which he tweeted, Shapps later accused the RMT of doing “lasting damage” to the rail sector and said that Network Rail, which manages the railway network across Great Britain, had tabled “a fair, legitimate pay offer” which he said amounted to an 8% pay rise over two years. “Employers have a pay rise on the table for RMT members in the train operating companies,” said Shapps, adding that the pay rise would not be put forward until the RMT agreed to reforms. The exchanges came on the third consecutive day of industrial action by transport workers this week. Network Rail and several train companies faced disruption on Saturday due to striking workers from the RMT and Transport Salaried Staffs’ Association (TSSA). Rail services will be substantially reduced, with only about a fifth – about 4,300 out of the usual 20,000 – running and half of the lines closed. Trains will operate only between about 7.30am and 6.30pm, while Sunday morning train services will also be affected. “I speak to thousands of our members every week, we consult at least 600 Network Rail reps on a weekly basis and we know exactly what the mood of our members is,” Lynch said. He added that management-grade members of the smaller TSSA union, who did vote to accept a pay deal, “have been bribed to break this strike”. “Our members are not going to be bribed, the offer is puny and they’re not ready to accept it.” Network Rail has said its latest pay offer is “good and fair” – and worth more than 5% – but was “conditional” on union members accepting “modernising reforms”. The company has said that unions have prevented other workers from accepting a deal. Andrew Haines, its chief executive, said: “It saddens me that we are again having to ask passengers to stay away from the railway due to unnecessary strike action, when we should be helping them enjoy their summers. “We have made a good and fair offer but, with the exception of our TSSA management grades who accepted the deal, our unions are refusing to let our employees have a say, and sadly that means more disruption on the rail network.” RMT has described the offer as a real terms pay cut, adding that reforms would lead to the loss of a third of frontline maintenance roles. Lynch said he would be talking to senior executives in the rail industry next week in an effort to “create solutions”. “Then we’ll decide whether we need to take more industrial action,” he added. Bus services in west and south-west London and parts of Surrey will also be disrupted on Saturday as the result of a strike on Friday and Saturday by bus drivers from Unite, in a separate dispute over pay. The transport secretary was accused of “advocating fire-and-rehire” tactics on Friday after warning that changes sought by rail companies could be imposed on workers. Grant Shapps told Sky News he could enact legislation referred to as “section 188” to force through some of the proposed measures. “If we can’t get this settled in the way that we are proposing, which is [asking unions], ‘Please put the deal to your membership’, then we will have to move to what is called section 188; it is the process of actually requiring these changes to go into place so it becomes mandated,” he said. But union leaders, including the TSSA general secretary, Manuel Cortes, said Shapps – who could lose his job when the next prime minister takes office – was “desperate” and “out of touch”. Speaking on Saturday, Lynch said public support for striking transport workers was growing. “I think the British public are fed up of being ripped off by this government and by corporate Britain, which have seen companies like BP and British Gas making massive profits while people are struggling to make a living,” he said. Wembley Stadium has advised those travelling to the capital to watch Coldplay on Saturday evening to check with their rail operator for the latest information.