Subject to planning approval, construction of the 10km road is expected to take place between October 2014 and May 2017.
The civil engineering work is estimated to be worth around £50m to each contractor, which signed the Early Contractor Involvement deal today.
Stockport Council, along with its partners Cheshire East and Manchester City councils have also today submitted a planning application for the £290m scheme.
The proposed relief road would bypass heavily-congested areas in Bramhall, Cheadle Hulme, Hazel Grove and Handforth.
The scope of the work includes completion of the detailed design, construction and defects liability maintenance of dual carriageway running from the A6 at Hazel Grove to the Ringway Road / Ringway Road West junction, incorporating an existing section of the A555.
It incorporates seven new and five improved junctions, four railway crossings and a parallel shared cycle and pedestrian path.
Cllr Sue Derbyshire, Leader of Stockport Council said: “These are significant steps in the delivery of the A6 to Manchester Airport Relief Road.
“Subject to planning approval, we look forward to working closely with Carillion and Morgan Sindall to deliver the scheme, which will improve the local economy and improve lives within Stockport by reducing congestion on local roads.”
Gary Crisp, contracts director for the Carillion Morgan Sindall joint venture, said: “Carillion and Morgan Sindall have an established joint venture history, having worked together successfully on the £209m Dishforth to Leeming contract to upgrade 22km of existing A1 which was completed on-time last year.”
The news follows the announcement earlier this month that the Department for Transport has awarded programme entry status for the scheme and a commitment of government funding, subject to completion of all the remaining statutory procedures.
The funding has been secured through a combination of £165 million of specific Department for Transport capital grant, £105 million of additional capital grant funding from the Government through the recently approved Greater Manchester Earn Back model, and £20 million of Local Transport Plan (LTP) funding.
