LNER said it is running as many services as it can, but is still affected by the Hitachi Class 800 trains that were taken out of service earlier this year after cracks were found in the chassis. The firm also said their trains are particularly busy on cross-border services.
Popular book website trainline is showing that trains from York to London at 12.32pm, 12.58pm, 1.33pm, 2.07pm, 2.32pm, 2.59pm, 3.33pm and 5.06pm are all sold out, with many of the other services only have space left in first class areas.
The train operator said no space remained available on its southbound east coast main line services via Edinburgh on Thursday and Friday, which are all reservation only.
Its website shows “no availability” on 24 trains on Thursday and 18 on Friday due to arrive before the kick off.
LNER tweeted: “Thursday 17th and Friday 18th June our services are SOLD OUT between #Edinburgh and #KingsCross due to the England Vs Scotland #Euro2020 match.
"Please only travel to the station if you have a reservation and travel plans in place. Our services are reservation only.”
A spokeswoman for LNER added: "Due to a number of Hitachi Class 800 trains being taken out of service for ongoing checks, some of our services have had to be cancelled or changed.
"Our train services are very busy, particularly on our cross-border services.
"We are contacting customers who have booked or reserved a seat with LNER directly on the affected services. We apologise for the inconvenience caused. All customers travelling with LNER require a seat reservation to travel and are advised to check our website lner.co.uk before arriving at the station."
London Mayor Sadiq Khan said fans who gathered in London to watch the game on Friday evening risked triggering another spike in Covid cases because of the highly-transmittable Delta variant.
The Scottish Government has also asked fans without tickets not to travel.
Fans travelling will also be unable to drink alcohol because of an LNER ban.
Mr Khan said the traditional gathering spot in Trafalgar Square would be out of bounds because it was being used as a ticketed fan zone for key workers, while pubs would have only limited space due to coronavirus restrictions.