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.
The Scottish Government has also asked fans without tickets not to travel.
Mr 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.
He has called on the Tartan Army to stay at home unless they have a “safe place” to watch the match.
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.”
The need for social distancing on trains and some services being cancelled by continuing checks for cracks on the underside of trains has reduced capacity.
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.
He said: “With Covid restrictions still in place on both sides of the Border, the best thing for fans without tickets to the game or a safe place to watch it is to enjoy the game from Scotland and not come to London.
"In an ideal world, I would welcome the Tartan Army to London for this match with open arms, but with Covid cases increasing, and with so much at stake as we fight this awful virus, I'm afraid that it just cannot be this time, so the best thing to do is not to come to London and instead enjoy the game at home."
Scottish culture minister Jenny Gilruth backed the plea.
She said: "I strongly urge fans to only travel if you have a ticket or a safe place to watch the match from."
UK sports minister Nigel Huddleston said the UK Government advised fans without tickets “to enjoy the match close to home and not travel to London unless you have somewhere already finalised to watch the game safely".
Just over 20,000 fans will be allowed inside Wembley for the match and less than 1,000 in the Trafalgar Square fan zone.
The office of the London mayor said it is not possible to arrange a separate screening in the city to accommodate large numbers of fans from Scotland under current Covid-19 restrictions.
LNER said: “LNER trains will be extremely busy across our route between June 17 and 20.
“This is due to large numbers of football fans travelling to and from London for the England v Scotland Euros 2021 game on June 18 and the cancellation of some services due to precautionary checks on a number of our trains.”