The Shropshire contractor's Bradford office will carry out the work on Neville Street, a tunnel linking the south side of the city with the centre.
The city council wants to reduce noise levels in the tunnel, which brings traffic in from the nearby M621 through Holbeck and Beeston, as well as improve its lighting and security. Footpaths, mainly used by local residents, will also be widened.
A sound absorbing material called QuietStone, which has been used in the Channel Tunnel and on the London Underground, will be installed and has been supplied Sound Absorption UK. Local firm Solar GB is supplying thousands of LED lights for the work which is due to finish this December.
The design team is led by Leeds firm Bauman Lyons Architects and includes Arup.