Network Rail bosses say they plan to hold more talks with union leaders today in a last-ditch attempt to avert the biggest strike on the railways for more than three decades.

More than 40,000 rail workers are due to walk out on Tuesday, Thursday and Saturday. Only about half of Britain’s rail network will be open on strike days with a very limited service on lines which are open.

One Network Rail source said: “We are in the room talking and there has been some movement. The chances are slim, but there is some hope.”

In his strongest comments yet over the looming strike action, the Labour leader Keir Starmer will say the strikes should not go ahead. He was to use a speech at a conference on Sunday to accuse ministers of “pouring petrol on the fire” to fuel a divisive dispute.

Network Rail has made some progress in talks planned to continue at its London headquarters on Sunday afternoon with the Rail, Maritime and Transport union (RMT). However, union bosses indicated on Saturday that there was almost no chance of a last-minute breakthrough in talks over pay, jobs and conditions.

Mick Lynch, the RMT general secretary, said “no viable settlements” had been tabled and the strikes would go ahead. He said thousands of jobs were being cut and workers faced below-inflation pay rises.

Network Rail has offered a 2% pay rise, backdated to January, plus two further rises during the year of 0.5% each linked to productivity and efficiency gains. Union negotiators want pay increases to match inflation.

Tim Shoveller, Network Rail’s chief negotiator, said: “I’m serious about trying to find a solution and work out a compromise that gives our people a decent pay rise, but it has to be affordable for taxpayers and fare payers.

“Our offers have so far been rejected, with union demands far from being affordable. We will continue to talk and to try and find a way through and avert this needless and damaging strike.” He said he was surprised the RMT was dismissing talks before they were finished.

The strike involves railway staff at Network Rail and 13 train-operating companies in the biggest dispute on the network since 1989.

Even if a last-minute deal was agreed over Network Rail workers, staff employed by the train companies would still go on strike. There will also be a 24-hour strike on London Underground on Tuesday. There will be no passenger services to stations across Britain if the strikes go ahead, including Penzance, Bournemouth, Hastings, Holyhead, Blackpool and Hull. The routes which are open will operate a very limited service from 7.30am to 6.30pm.

Passengers are being advised by the train companies to travel only if necessary, with most commuters likely to work at home on strike days.

Glastonbury festival-goers are among those likely to be affected.

With fine weather forecast for many areas this week, the action is set to disrupt several summer events, including the Glastonbury festival, the British Athletics Championships in Manchester and the third Test match between England and New Zealand at Headingley, Leeds.

The industry group UK Hospitality warned on Saturday the rail strikes were likely to cost the tourism, leisure and theatre industries more than £1bn.

Kate Nicholls, chief executive of industry group UK Hospitality, told Times Radio that tourism and leisure businesses which were already fragile after pandemic lockdowns would take a “big hit”.

Some A-level and GCSE examinations are being held on the days of the action. The Department for Education has said it did not expect any examinations to be rescheduled because of the action and advised pupils and teachers who travel by train to consider alternative travel arrangements.

Grant Shapps, the transport secretary, said the government had repeatedly urged the RMT to resolve the dispute at the negotiating table. He said on Saturday: “Sadly they have ignored these requests, and we are now on the cusp of major disruption which will cause misery for people right across the country.

“Many people who do not get paid if they can’t get to work face losing money at a time they simply can’t afford to.

“The RMT is punishing millions of innocent people, instead of calmly discussing the sensible and necessary reforms we need to make in order to protect our rail network.”

RMT bosses must negotiate with Network Rail and train company bosses over the pay deals, but says the Treasury is “calling the shots”. Shapps last week rebuffed a call to meet union bosses, saying the talks were rightly being conducted by the employers.

The Observer understands Network Rail has been given a negotiating mandate and framework by the government. Once any deal is agreed, it is likely a similar mandate will then be given to the train companies.

Starmer is due to use a speech at the Labour Local Government Association annual conference in Coventry to accuse ministers of preferring to use strike action for political attacks, effectively encouraging them to go ahead.

“Businesses will struggle with freight. School exams will be hard to get to. Hospital appointments missed,” Starmer will say. “That’s why I have said the strikes should not go ahead. But here’s the truth. Boris Johnson and Grant Shapps want the strikes to go ahead. They want the country to grind to a halt so they can feed off the division.

“Instead of spending their time this week around the negotiating table, they are designing attack ads. Instead of grown-up conversations to take the heat out of the situation, they are pouring petrol on the fire. Instead of bringing people together in the national interest, they are stoking division in their political interest.”

The Rail Delivery Group (RDG), which represents the rail industry, says the government committed more than £16bn to keep the railways running during the pandemic and reforms were required to attract passengers back on the network.

An RDG spokesperson said: “No one wins in the event of a strike. Working with Network Rail, our plan is to keep as many services running as possible, but significant disruption will be inevitable and some parts of the network will not have a service, so passengers should plan their journeys carefully and check their train times.”

Which are the sectors likely to be affected by strike action?

Transport

A second rail union, Transport Salaried Staffs’ Association, has served notice to ballot 6,000 staff at Network Rail for industrial action for a pay increase to reflect the cost of living crisis. In the event of a yes vote, strike action could take place from Monday 25 July, the week the Commonwealth games start in Birmingham.

Local government

The three local government unions representing 1.4 million council workers have made a joint submission to local government for pay rises at the rate of the retail prices index, currently at 11.1% or a £2,000 pay increase, whichever is the greater.

Civil servants

The Public and Commercial Services Union, the largest trade union in the civil service, is set to ballot its members on industrial action later this year over 2022-23 pay proposals. It is calling for a national pay rise of 10%.

Teachers

The National Education Union, which represents 460,000 members, has committed to a ballot in the summer or autumn term if they do not secure pay increases for teachers across the board of at least 8%.

Medical staff

Junior doctors in England say their pay has declined 22% in real teams since 2008-09. The doctors’ union the British Medical Association says it will ballot for industrial action by early 2023 at the latest if the demand to restore junior doctors’ pay is not met. The Royal College of Nursing is demanding a pay award for nurses of 5% above inflation, but the government has requested to cap any pay increases for 2022-23 at 3%.

Barristers

The results of a ballot of 2,400 criminal barristers will be announced tomorrow on strike action. Criminal trials could be halted with possible walk outs from Monday 27 June.