Work on the £75m project is expected to start later this year, subject to planning consent and is expected to take approximately 20 months to complete.
If planning permission is granted, the proposals would see the capacity of a new main stand rise by approximately 8,300, taking the overall capacity of the Anfield stadium to around 54,000.
Carillion will keep as much as possible of the existing main stand in operational use throughout the contract.
Chief executive Richard Howson said: "I believe that being selected for this prestigious project reflects Carillion's reputation for high standards of quality, health & safety and sustainability. We look forward to working closely with Liverpool Football Club to create new world-class facilities at its famous Anfield stadium."