Anderson, 23, is currently at the World Cup with England and has already completed a medical in Kansas.
The formalities of the move will be finalised upon his return to England.
"Manchester City and Nottingham Forest have reached an agreement for the transfer of Elliot Anderson.
"In the meantime, everyone at Manchester City wishes Elliot and the England squad the very best of luck in their World Cup campaign and we look forward to welcoming him to Manchester in due course."
Neither club has disclosed the fee or length of contract involved in the deal.
However, well-informed British media report that City will pay in the region of £116m for the midfielder. That fee would take Anderson beyond Jack Grealish as the club's most expensive signing.
Anderson will link up with his new Manchester City teammates once his involvement with England at the World Cup comes to an end.