Initially, Carillion will advise on how to best to carry out a £10m programme of extensive concrete repairs to the roof and walls of both tunnel bores.
The job also involves special strengthening of the tunnel supporting the Leeds General Infirmary buildings for potential vehicle impact.
The Woodhouse Tunnel is suffering severe concrete damage and deterioration from extensive chloride attack caused by winter de-icing salts.
The damage is so severe that the structural capacity of components is now being affected.
Construction repairs will run for 20 months.