The Aberdeen-based firm has run passenger trains on the route with its French partner Keolis since 2004.
Keolis and its new bidding partner Go-Ahead are also on the shortlist, along with Stagecoach Group, which jointly runs the West Coast Main Line.
The TransPennine Express runs from Manchester to Glasgow, Edinburgh and Newcastle.
The finalists will now make full bids for the route, before the Government makes a decision on the new operator and length of the contract by October 2015.
"We look forward to reviewing the contract details and submitting a compelling and value for money bid which meets the needs of taxpayers as well as customers and businesses along the route," said FirstGroup's UK rail director Vernon Barker.