The firm was named preferred bidder of for the second stage of the ground's updated Masterplan.
The two existing stands will be placed with new three-tier stands.
Cricket Club (MCC) also confirmed it has been granted planning permission for the redevelopment work.
Chief Executive & Secretary, Guy Lavender, said: "MCC is delighted to receive planning permission for the redevelopment of the Compton and Edrich stands. We are excited to be commencing another transformational development at Lord's."
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The new designs include integrated catering and WC facilities, and an improved spectator experience. The seating capacity will be increased by around 2,500, accommodating 11,500 spectators overall, as well as wheelchair-accessible spaces and lift access at all levels.
Construction is due to commence after the completion of this summer's major matches in August 2019, with the new stands' seats due to be ready for use during the 2020 season.
Fully fitted-out facilities are scheduled to be completed for the summer of 2021.
(CM/MH)
