The revised offer includes higher rest day working payments and other changes to working arrangements.
This includes more organised allocation of work on rest days, a union spokesman said.
The RMT said train managers will vote on the offer in a referendum that finishes on 17 December.
Train managers usually work a 41-hour week but due to staff shortages can be asked to work on their days off.
The RMT had said that this situation was “unacceptable”.
About 300 train managers had been expected to join the walkout, limiting train services on those three days.
Avanti runs services between London and Glasgow and Edinburgh, plus to stations including Birmingham, Liverpool and Manchester.
Avanti West Coast said it was “pleased that the RMT have decided to suspend these strikes which would have caused major disruption to our services”.
“This will come as a relief to our customers who were making travel plans over the Christmas period,” it said, adding that it remains open to discussions with the union.
