It came as cross-Border train operators such as LNER and Avanti West Coast were discussing possible extra services to meet demand during the 23-27 Christmas travel period announced by the Scottish and UK governments.
The operators warned passengers last week some of their trains would sell out or may be “exceptionally busy”.
Avanti has since brought forward the start of bookings from Tuesday to yesterday and will remove peak fares from December 18.
There is compulsory seat reservations on LNER trains.
Train capacity has been reduced to as little as 30 per cent because of the need for social distancing.
Network Rail confirmed it was discussing with operators the potential for extra trains being laid on.
RMT general secretary Mick Cash called for urgent talks with Network Rail chair Sir Peter Hendy, who has been tasked by the UK Government to check all Christmas travel plans are “smooth and safe”.
He called for “increased police enforcement and penalties for passengers not sticking to the rules” and “early action to limit passenger number to a safe level to ensure numbers are agreed and known in advance to avoid last minute chaos and overcrowding”.
Mr Cash said: "We know that the [UK] Government has a track record of operating on the hoof throughout this pandemic and just three weeks away we cannot allow Christmas travel plans to fall foul of that slap dash approach.
"Operational and safety arrangements need to be discussed and agreed with the unions as a matter of urgency."
UK Transport Secretary Grant Shapps said: “For those that choose to form a Christmas bubble, we’re lifting travel restrictions across the UK for five days for the purposes of seeing that bubble.
“Before you travel, plan your journey very carefully, and where possible book well in advance.
"Everyone must also follow the clear guidance to keep you, fellow passengers and staff safe.
“As some advance tickets go on sale and people begin to plan their journeys, we are closely assessing demand on the network and have already taken actions to minimise potential disruption.
“We are currently developing a plan focused on tackling disruption - including running longer trains and relaxing rules to allow more types of coaches to run - and will publish further details once demand is clearer.”
The UK Department for Transport said it had asked Sir Peter “to rigorously scrutinise the winter preparedness plans of all transport operators to ensure journeys are smooth and safe”.
An Avanti West Coast spokesperson said: “From December 18, we’re moving all our services to off-peak fares to help the Christmas getaway – making it cheaper for everyone to see their loved ones over the festive season.
“We expect services to be busier over the Christmas period in comparison to the last couple of months.
“We’re looking closely at putting on extra services with Network Rail and the Department for Transport.”
Robert Nisbet, director of nations and regions for the Rail Delivery Group industry body, said: “Rail companies are working around the clock with government to ensure passengers can travel with confidence over the festive period.”