Members of the Rail, Maritime and Transport (RMT) union walked out on Saturday for the third time this week, with little sign of a breakthrough in discussions between the union and rail operators.
Only a fifth of services are running and half of the lines are closed. Passengers have been warned by rail operators that they should only travel by train if necessary and to check their journey in advance.
Mr Lynch said the union has not ruled out further strikes given there is little sign of a breakthrough to the deadlocked row.
Speaking to the PA news agency at a picket line outside Euston Station in central London on Saturday morning, he said: “We’re not ruling out strikes but we have not put down any dates for any strike action.
“We’re going to review with our national executive next week, who have been all the way round the country this week on the picket line, so we’re all going to get together the leadership of the union and see where we are.
“We are not going to name dates immediately and we’re going to continue working constructively with the companies to strike a deal, but that is a really steep challenge at the moment because of the agenda they’ve got and the effects they want on our members.
“Strike action’s not ruled out and it will have to take place if we do not get a deal, but we’re hoping that we can get a deal and we get some compromise.
“What we try to do is have the most effective strike action if it needs to take place. We’re not just pretending. It’s got to be a coherent and effective strike action because we don’t want to waste our members’ energy on something that doesn’t work. We’ll review that and see what we need to do if we need to take that action.”
Mr Lynch also hit out at the Government, saying many ministers have “never done a hand’s turn”.
He said: “They’re not using the system that they want and they’ve never worked in this type of work.
“Many of them have never run a business, but they’ve also never worked on the tools. As my mother would say, they’ve never done a hand’s turn.
“It’s quite odd. The people who are running this country are brought up on a diet of Latin and Greek and our members are brought up on a diet of getting up at ungodly times to run the transport system. I think there’s a bit of disconnect there.
“If we had people who were used to doing work we might get a better deal out of them.”
Later, Mr Lynch spoke at an RMT rally outside King’s Cross St Pancras station, alongside Labour MP Diane Abbott and actor Rob Delaney.
Several hundred people gathered with many holding placards reading slogans like: “Tories out”, “cut profits not pay” and “support the railway workers strike.”
Meanwhile songs like the “Internationale” workers anthem and “Solidarity Forever” played from speakers on a makeshift podium.
Ms Abbott told the crowds: “We’ve seen the private sector ripping off the public purse, bearing down on wages and conditions, undermining job security and we’ve seen a public sector that is increasingly demoralised.
“But the RMT is drawing a line in the sand against all this and offering leadership to working people everywhere and that is why it is so important that the RMT wins this dispute.”
Meanwhile, Delaney told the crowds the strike was “inspiring” and called RMT’s action the “tip of the spear”.
“I offer my unwavering support and solidarity,” he said.
Speaking to PA after his speech, the actor said: “I ride trains like you ride trains and I would like the people who clean them to be fairly paid, or not have a pay cut in real terms or have their job removed.”
He added: “We’re not asking for crazy things, these people are asking to be able to heat their homes, people who have jobs want to be able to heat their homes in winter.
He added that he hopes the dispute gets resolved because those striking should not be out protesting in King’s Cross on a Saturday in June but should be outside or “on the beach”.
It comes after Prime Minister Boris Johnson said the public has a right to expect reforms to rail services.
He told Sky News: “I would say, given the circumstances we’re in, I think what we want to see is reform and improvement in the way the railways work, and modernisation.
“When you’ve got a 25% fall in ridership, which we’ve got at the moment, we’ve got the Government putting billions and billion (into it).
“We’re putting more into the railways than any previous government.
“I think the travelling public has a right to expect some basic reforms, like with ticket offices, like with walking time, and some of these other practices that really nobody defends except the union leaders.”