Costerton is set to replace the agency's EMEA chairman and CEO Andy Laurence.
After five years in the job, Laurence is moving into a new role of senior corporate counsel and chairman of the firm's worldwide corporate practice committee.
Richard Millar, who is currently head of marketing communications at the agency, is due to replace Costerton as UK CEO.
H&K is to make the changes to its senior management from January next year.
Laurence joined the company in 1989. He has held a range of senior roles, including chairman of European practices, head of corporate practice EMEA, UK deputy chairman and UK CEO.
Costerton joined H&K in 1999 as head of technology practice EMEA before being promoted to the top UK job in 2006.
Millar, who spent four years at H&K in the 1990s, rejoined the agency in 2004, after eight years in senior marketing and management roles, latterly as Habitat's UK CEO.