MediaMonks is in pole position to be appointed to handle Braun's digital advertising account.
MediaMonks, which Sir Martin Sorrell acquired soon after resigning from WPP, was pitching against WPP’s Possible for the Procter & Gamble brand.
Campaign understands that Possible is now out of the running.
MediaMonks directed requests for comment to P&G, which left repeated requests for comment unanswered.
WPP’s Grey handles the Braun global creative business, after it picked up the account from Omnicom’s BBDO in 2015.
Sorrell, who left WPP suddenly in April after 32 years, set up a new company called S4 Capital in May. He acquired MediaMonks for €300m (£266m) in July, seeing off competition from WPP.