Sources close to the project said confirmation of the award is imminent.
Earlier this week HS2 handed-out £3bn of track contracts with Ferrovial BAM and Colas Rail among the big winners.
A project source told the Enquirer: “The tunnel fit-out deal is another major package on the project that everyone has been waiting to hear about and the word is that Costain has landed it.”
Costain declined to comment.