MediaCom is understood to have beaten Initiative, M/SIX and the Publicis Groupe media agency ZenithOptimedia to the business.
In January, Tesco moved its creative account into Bartle Bogle Hegarty without a pitch. The creative business was previously handled by Wieden & Kennedy.
The review follows the arrival of Dave Lewis, the former president of personal care at Unilever, as the chief executive of Tesco in September 2014.
Tesco and Initiative had not responded to a request for comment by the time of publication.
Update on 30 April - Karen Blackett, chief executive of MediaCom, said: "Tesco are a brand we've admired and an organisation we've wanted to work with for years. We're so glad we've been given the opportunity. It's going to be great to partner with BBH and Blue Rubicon. We can't wait to get stuck in."