Train drivers have been offered a 4% pay rise for two years in a row by the body that represents rail companies in a bid to end strike action.
The Rail Delivery Group (RDG) has made its first offer to Aslef, the union for train drivers, after several strikes.
The deal includes a backdated pay rise of 4% for 2022 and a 4% increase this year, but it also hinges on changes to working practices.
Aslef told the BBC its officials had not seen the offer yet.
But its General Secretary Mick Whelan has previously said that the union is "chasing a pay rise that at least puts a dent" in prices, which are rising at their fastest rate in 40 years.
The RDG said it sent its offer to Aslef mid-afternoon on Friday after drivers at 15 train companies walked out on Thursday, leaving some operators unable to run any trains.
The action by drivers comes as other rail workers, such as guards and signalling staff in the RMT union, continued a series of large-scale strikes.
Saturday marks the second day of a 48-hour walkout by tens of thousands of RMT members, with only around 20% of services on the rail network expected to run.
Meetings between the rail minister, industry representatives and union leaders will take place on Monday.
The RDG said its offer, the first made to Aslef during to the dispute, included having no compulsory redundancies until the end of March next year.
If accepted by members, it would mean the average salary for a driver would increase from £60,000 per year to £65,000 by the end of 2023. Ten years ago it was £44,985.
It is understood the deal is contingent on what the the group has described as "common sense, vital and long overdue changes to working arrangements across the industry", which the RDG argues will deliver a more reliable service to passengers.
One of the conditions the RDG has outlined is employers taking control of things like staff, work and training schedules, meaning bosses would not need to agree rotas or train routes with unions.
Other proposed changes include:
A new protocol which would mean drivers who are rostered to work a Sunday shift are committed to doing that unless they can get cover
Reducing time it takes to train drivers through "better use of technology", such as using simulators
Managers deemed as "competent" being allowed to drive trains during disruption on the network
Introducing part-time and flexible working
The RDG said the changes were "vital in a post-Covid world", which has seen leisure travel recover more quickly than commuting has and companies struggling with a big hole in their finances.
It said the offer also looks to "address many inefficient and arcane practices", using a policy of extra payment for the use of technology such as tablets and smartphones as an example.
Steve Montgomery, chair of the Rail Delivery Group, said: "This is a fair and affordable offer in challenging times, providing a significant uplift in salary for train drivers".
He added that the deal would bring in "common-sense" reforms that would allow the railway to adapt to changing travel patterns.
"Instead of staging yet more damaging strike action and holding back changes that will improve services, we urge Aslef to work with us," he said.
If Aslef accepted this offer, it would still need to go down to local level and agreed at each of the 15 train companies where it represents drivers.
Last month, the RMT union rejected an offer from the Rail Delivery Group, which included the same pay rise conditional on a list of changes to working practices.
The RMT objected to some of those, particularly the expansion of driver-only operation, where drivers operate the doors on all carriages.
It is understood this is an area under discussion, as all sides look for a way forward in the ongoing dispute