It is thought that he has become disillusioned with the progress of the WPP-owned Y&R in the past year.
A source close to Costa also said that his relationship with Hamish McLennan, the chief executive of Y&R, has soured recently.
However, Costa denied that he had left the company. He said: "I have not resigned from Y&R and have no plans to. That is the truth."
WPP is known for taking a strict line with outgoing executives both in terms of what they can talk about and when they can return to work.
Y&R and WPP refused to comment on the story.