His departure follows the shock exit in December last year of his partner Robert Campbell, the former McCann executive creative director and the agency's vice-president for Europe, the Middle East and Africa.
Reddy, 48, was poached by McCann in May 2004 from his position as head of art at DDB London. He was charged with leading an overhaul of McCann's visual output, including the agency's image.
He worked alongside Campbell and Frank Lieberman, who was lured out of retirement to take up the position of executive head of TV early in 2004.
BBH does not currently have a head of art. The agency's executive creative director, John O'Keeffe, said: "Over the past 15 years, the record has shown that Mark is the best art director there is. He is joining the right company at the right time."
During his time at DDB, Reddy was credited with helping to develop the agency's print credentials - it won the Press Grand Prix with its "cops" execution for the Volkswagen Polo at the 2004 Cannes Lions.
Reddy began his career in 1975 at Ogilvy Benson & Mather. In 1997, he moved to the London Advertising Partnership (later Colmans) and also had spells at Holmes Knight Ritchie and Saatchi & Saatchi before joining Doyle Dane Bernbach in 1986.