High-speed services between London, the West of England and Wales were affected last week after Rail, Maritime and Transport (RMT) members walked out.
First Great Western (FGW) said it made a "substantially revised offer" but a sticking point remained - it wants drivers to have control over doors.
RMT is yet to comment on the new offer.
An FGW spokesman said: "We don't believe there are any fundamental differences of opinion between us and the RMT on issues relating to engineering, station and dispatch staff...
"The key issue that remains is that we want the driver to close the doors - which is key to our ability to run the faster, more frequent services customers should expect from a 21st Century railway.
"To think that over 2,000 staff could be asked to take more strike action and further inconvenience our customers when all but one of the issues has been dealt with just doesn't make sense."
According to the union, about 2,000 members of staff went on strike across the FGW network.
In his latest open letter to customers, FGW managing director Mark Hopwood said there could be further strikes.
He wrote: "We have made very clear our commitments on job security and the customer experience.
"There remains, however, the possibility that we will face more strikes and we are training more of our people in more tasks so that we can run as comprehensive a service as possible."
The first Hitachi trains will run on the Great Western main line from 2017 and the East Coast main line from 2018.
The new trains are being introduced under the Intercity Express Programme (IEP) on routes between London Paddington and Oxford, Bristol and South Wales.
Train operators say they have failed to agree with transport workers on the finer points of a deal following a 48-hour strike.