The proposed relief road would bypass heavily-congested areas in Bramhall, Cheadle Hulme, Hazel Grove and Handforth.
Stockport Council aims to get a contractor on board by next August to work up design details for construction work to start in October 2014.
A business case for the scheme will be submitted shortly to the Department for Transport.
The scope of the work will include 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.
Stockport Council, along with its partners Cheshire East and Manchester City councils, are consulting on the road scheme in two phases.
The first phase of consultation is now underway and runs until 25 January 2013 and is looking for views on junction options to help determine a preferred scheme.
A second phase will follow in spring 2013 when the three councils will seek views on the preferred scheme, prior to the submission of a planning application in summer 2013.
Interested firms can register at the Chest Portal or contact Andrea Boulton, tel: 01614744103.