Cook had been the subject of internal investigation after sending an offensive e-mail to the mother of defender Nedum Onuoha.
The club have issued a statement confirming the board had accepted Cook's resignation.
It was reported that Cook initially claimed his e-mail had been hacked into but has now admitted to an "error of judgement".
Cook has also been criticised in the past for a number of public relations gaffes.
Cook said: "I am privileged to have held my position at Manchester City Football Club and to have experienced the opportunities that it has presented.
"The privilege is in part offset however by the significant personal focus which has at times, detracted from the magnificent achievements of those working at the football club.
"It is that factor, together with my error of judgement in this matter that has prompted me to reach this decision, which I believe is in the best interests of the football club."