Rival bidders told the Enquirer the job has gone to McLaren who will start on site shortly.
The deal was also confirmed by development partner Stoford.
Work is expected to start on site shortly and take 12 months to complete.
The 1.2million square feet warehouse will be built at the Central Park logistics centre in Avonmouth and will be the size of 15 Wembley Stadiums.
Chris Dawson, founder of The Range, said: “This distribution centre is a big step in the expansion plans that I have for the business; it’s non-stop for us.
“When the warehouse is up and fully operational, it will act as a training hub for smaller distribution centres around the country.”
Stoford has also bought another 20 acre site at Central Park.
The firm plans a £43m speculative development of three warehouse and logistics units which will be built by main contractor Winvic Construction.