Cook, the chairman and managing director of Burnett, has been with the company for 17 years through its different incarnations.
Bruce Haines, Burnett's group chief executive, explained that he is now looking at other roles for Cook within the Publicis Groupe.
"We are still in the process of discussing Barry's future ambitions.
It is very much my hope that we will be able to offer him a new and different role that will both inspire him and, more selfishly, keep his talent for the group," he said.
"Barry will be with us for as long as it takes me to make decisions about the agency's future management structure," he added.
As the managing director at D'Arcy, Cook was involved in the company's merger with Burnett following its acquisition by the Publicis Groupe in 2002.
The merger saw Haines and Cook teaming up to share the responsibility of running the new agency. However, Burnett has since failed to win significant new business, and has failed to make a creative impact.
Cook's departure follows that of Paul Shearer, the joint executive creative director. Shearer quit the agency in May without a job to go to, leaving Jim Thornton as its sole creative head.
Cook joined Yellowhammer in 1988. The agency was rolled into D'Arcy Masius Benton & Bowles in 1992. He was made the managing director at DMB&B in 1996. Previously, he worked at Leagas Delaney and Braybrooks and Garett and he began his career on the client side as a salesman at Procter & Gamble.
Cook said: "What most people aren't aware of is that I joined this company 17 years ago ... albeit that it was then called Yellowhammer. During that time, I have worked under four different brand names and I have seen the company's status and fortunes change dramatically.
"I've had some great jobs, worked on some great brands and, most importantly, worked with some wonderful people."
Of his departure he added: "For the first time in a long time I think I need to move on to find a challenge. Importantly, I leave Leo Burnett in good hands and during my notice period I will do everything possible to ensure a smooth handover."
Haines paid tribute to Cook's work for the agency: "Barry has worked long and hard for our company and we owe him a great debt. He has provided first D'Arcy and now Burnett with first-class leadership and the agency will now build on that legacy."