Workers were due to walk out from 5pm on Thursday 4 June for 24 hours and then again from 5pm on Tuesday 9 June for 48 hours.
"After four days of intensive talks, Acas has helped Network Rail, RMT, TSSA and Unite formulate a set of revised proposals that the trades unions will now take away to consider," said a spokesman for Acas.
"Recognising this, the RMT has agreed to suspend the industrial action planned for this week and next."
RMT general secretary Mick Cash said the revised offer enabled the union to suspend the industrial action while it consulted in full on the details of the package.
Mark Carne, Network Rail chief executive, said he was "very pleased" that the strike had been called off, adding that he hoped the union would agree to the new deal.
According to Sky News, the new pay offer is believed to comprise a two per cent pay rise for 2015 and a pay rise linked to RPI inflation next year, as well as a "job security package".
It is the second time a UK-wide rail strike has been averted in the space of two weeks. A walkout planned for the Monday of May bank holiday was also called off.
All three strikes were expected to cause widespread disruption to the country's entire rail network.
Rail strike: union rejects new deal and plans double strike
29 May 2015
After calling off the bank holiday rail strike, unions are now threatening to hold not one but two strikes next month.
Rail, Maritime and Transport (RMT) union members at Network Rail are planning to walk out from 5pm on Thursday 4 June for 24 hours and then again from 5pm on Tuesday 9 June for 48 hours.
Overtime will also be banned from Saturday 6 June to Friday 12 June.
Staff were due to strike earlier this week on bank holiday Monday but suspended the walkout after Network Rail offered them a new pay deal.
This has since been rejected, although RMT said it would remain "available for talks". The union has called on Network Rail to improve its offer.
Transport Secretary Patrick McLoughlin has warned that "millions of hardworking people" would be disrupted by the strike and insisted that RMT members already get a fair deal.
"It is very disappointing that RMT has now rejected a deal delivered through Acas talks that the union's leadership agreed was a reasonable offer," he told the BBC. "The government will do everything it can to help keep people and goods moving during the strike."
Mick Cash, general secretary of RMT, said his members had been left with "no option" but to start industrial action again.
Network Rail initially offered a four-year deal including a one-off £500 bonus but no pay rise this year, with future rises limited to RPI inflation until 2019. Employers had also promised to make no compulsory redundancies before December 2016 for nearly 16,000 staff.
The new offer spans just two years, with a one per cent rise this year and a rise of around 1.4 per cent next year, with no compulsory redundancies for the next two years.
Mark Carne, Network Rail's chief executive, said it was "clearly unacceptable for the RMT to massively disrupt the travelling public with strike action when we are ready to continue talks".
Rail strike called off after new pay deal
21 May 2015
A national rail strike that had been expected to leave fewer than one train in ten running on Monday and Tuesday has been called off after a new pay deal was tabled by Network Rail.
"Following the Acas talks, RMT has received a revised offer that enables us to suspend the planned industrial action while we consult in full with our Network Rail representatives," said Mick Cash, the union's general secretary.
The TSSA union, which was also planning strike action, said earlier today that it was also abandoning its walkout.
Rail passengers have been warned to prepare for widespread disruption at the end of the bank holiday weekend.
South West Trains had cancelled all trains on Tuesday and Virgin was not planning to run any services on either day. Some trains were expected to run on the East Coast mainline, but none would cross the border between England and Scotland.
The dispute arose when rail workers rejected a four-year deal in which Network Rail offered a one-off £500 bonus but no pay rise this year, with future rises limited to RPI inflation until 2019. The employers had also promised to make no compulsory redundancies before December 2016 for nearly 16,000 staff.
On a 60 per cent turnout of the union's 16,000 members, 80 per cent voted to strike, while 92 per cent voted for action short of a strike.
The history of Donald Trump's election conspiracy theories
Two ancient cities have been discovered along the Silk Road
'People shouldn't have to share the road with impaired drivers'
Driving home for Christmas: 20m UK car journeys expected this week
Rail strikes: whose side is the public on?
Mick Lynch: the veteran trade unionist leading rail walkouts
Rail strikes: is Britain on track for a ‘summer of discontent’?
‘See it. Say it. Sorted’: is it the end of the line for train announcements?
UK to bring in airport Covid tests for arrivals
UK records biggest jump in transport use since pandemic began
Coronavirus: what are the odds of catching Covid-19 on a plane?
