The union in dispute with South Western Railway (SWR) - over the role of guards - has written six points it wants addressed to reach a deal.
Last week, Rail, Maritime and Transport (RMT) union members started 27 days of strikes - affecting tens of thousands of commuters.
The dispute has now been running for two years, the disagreement is surrounding the duties of guards on 90 new trains.
To make journey times more efficient, SWR wants drivers to operate the passenger doors at every stop.
Union members instead argue that - for passenger safety - guards should decide when to close the doors.
3/3 ...to this dispute.— RMT (@RMTunion) December 9, 2019
" If the company can give us an assurance that they accept the principles we have outlined that would enable us to suspend the current action."
SWR runs services between London Waterloo and Portsmouth, Southampton, Bournemouth and Reading - as well as North Hampshire and South West Surrey.
RMT General Secretary Mick Cash said: "We are literally three seconds away from a deal that would achieve both our objectives and the company's.
"I would appeal to SWR to get out of their bunker and back into ACAS with the union to bolt down a settlement.
"A negotiated solution to this dispute which would cost the company nothing and meet RMT's objectives of protecting safety and accessibility is within grasp."
In response, South Western Railway told The Breeze it welcomed RMT's proposal, but has not yet received the plan directly.
SWR told us it needs to know what is different about the RMT's latest publication
A spokesperson said: "We actually agree with most of the RMT's points, but on a busy metro network like ours every second counts and we cannot have unnecessary and inefficient practices that increase delays for passengers.
"Only by doing this can we free up guards to offer the customer assistance the RMT describes, making our trains more accessible, and providing better customer service to everyone."
The industrial action is scheduled to finish on 1st January.