It warned the further shutdowns will be on a similar scale to this weekend’s closure of King’s Cross station, when no trains will run between there and the Scottish capital.
LNER and Network Rail said they were unable to rule out a repeat over the August Bank holiday weekend in 2020 and/or 2021.
Edinburgh Festival and Fringe chiefs have claimed this weekend’s disruption will be “totally disastrous”.
Closures of the east coast main line are part of a £1.2 billion upgrade for faster and more frequent trains.
It will enable LNER to run services every half hour between Edinburgh and London, with journey times for some being cut to four hours.
LNER said it had taken the unprecedented step of urging passengers not to travel between the capitals on Saturday or Sunday because its trains will only be able to go as far south as Peterborough.
Passenger numbers are expected to be cut by two thirds during one of the only weekend closures of King’s Cross in its history, to renew and streamline tracks.
There will also be very restricted services on Monday.
Virgin Trains said its services between London Euston and Edinburgh via the west coast main line would be full, and other passengers will have to travel via Glasgow.
LNER safety and operations director Warrick Dent said there would be between four and six more major disruptions on a par with this weekend’s, with some likely over Christmas periods.
He said: “There is never a good time, so it is a case of finding the least worst time for the work.
“There will be more weekends similar to this one in 2020 and 2021, largely because of work at King’s Cross.”
Mr Dent said fewer people travelled over bank holiday weekends, when business passengers were not affected.
David Sidebottom, a director of passenger watchdog Transport Focus, said: “Investment in maintenance and improvement is necessary, and people understand that.
“But our research is clear: passengers want to be kept on the train wherever possible, they want to know before buying a ticket if part of the journey will be by bus, and they want plenty of staff on hand to signpost and to help.”