Cheshire West and Chester Council has chosen the firm to take the £13.5m project forward.
Graham is believed to have fought-off competition from Balfour, Kier and Galliford Try for the job.
Graham will now work alongside scheme architects Austin Smith Lord, transport engineering advisers Mott MacDonald and council officers to prepare detailed implementation plans.
Work is expected to begin on site in mid-September and the new facility is due to open in early 2017.
Councillor Brian Clarke, Cabinet Member for Economic Development and Infrastructure, said: “We look forward to working in partnership with John Graham Construction to deliver a first class, modern transport facility that will improve the experience of bus users and encourage more people to take advantage of public transport.
“The new interchange at Gorse Stacks will bring about a major improvement in the quality of Chester’s offer for residents and visitors alike.”
Gary Hughes, Regional Director of John Graham Construction, said: “We are delighted to have been appointed as the Council’s construction partner and look forward to delivering a high quality bus interchange for the city of Chester.”