It caused electricity to jump onto adjacent wires and cause damage, Network Rail said.
The incident between Lockerbie and Carlisle about 10:30am is expected to keep the west coast main line to Glasgow closed for the rest of the day.
Avant West Coast, which operates trains on the line between Glasgow, Edinburgh and London, was among train operators affected.
Around 40 of its services were disrupted.
TransPennine Express trains on the line between the two Scottish cities, Manchester and Liverpool were also hit.
The bird strike added to earlier disruption on the line for two hours this morning caused by a dip in the track due to loose ballast near Thankerton in South Lanarkshire.
Avanti advised passengers: "Due to damage to the overhead electric wires at Carlisle, all Glasgow and Edinburgh lines are blocked.
"Disruption is expected until the end of the day."
It said tickets could be used on CrossCountry, LNER and ScotRail trains via Edinburgh instead.
TransPennine Express said: "Services from Scotland towards Manchester/Manchester Airport are currently terminating and starting from Carlisle.
"Services between Liverpool Lime Street and Scotland are terminating/starting from Preston."
Network Rail said bird strikes caused occasional disruption.
A spokesman said: "It causes electricity to arc onto catenary wires.
"Engineers are making repairs."