The line was closed on Monday morning because of strong winds and heavy rain across the east coast of Scotland and the north-east of England.
Affected operators include London North Eastern Railway, CrossCountry and TransPennine Express.
Network Rail said it was carrying out safety checks in a bid to reopen the line.
These include monitoring overhead electric wires to ensure they have not been damaged in the severe weather, and running trains to check for debris on tracks.
The organisation said its staff have faced “an exceptionally challenging weekend with two significant storms bringing severe weather”.
Storm Malik caused disruption on Saturday, before the arrival of Storm Corrie on Sunday.
Much of the ScotRail network has been affected by the severe weather.
The operator warned passengers on Monday that it has only been able to arrange “very limited replacement transport”.
ScotRail withdrew all of its services on Sunday night in an effort to “protect passengers and railway staff”.
Network Rail Scotland, which said that “all parts of the railway are working together”, added that any other trains that ran on Sunday night had a maximum speed of 40mph.
This morning ScotRail tweeted: “After the storm Network Rail will have to complete safety checks on some routes before we can open them back up for passenger services. Some of these routes are already complete but some can only be done in daylight.
"Due to very limited replacement transport being secured, passengers are advised to consider making their own alternative arrangements this morning.
"To find out information on specific services or the replacement buses and ticket acceptance we have managed to secure, please visit our JourneyCheck page.”
Network Rail also shared an image showing trees and debris on the line between Aberdeen and Inverurie, tweeting: “This is just one example of what we have had to deal with this morning due to #StormCorrie. Our teams are continuing to clear the lines at the moment”