The 23-year-old will join City after an impressive spell with Forest that saw him break into the England national team and become regarded as one of the Premier League's best midfielders.
Anderson arrives at the Etihad Stadium for a reported fee of £116 million, paid upfront without instalments - making him the fifth-most expensive transfer of all time.
The formalities around his deal will be finalised once England's World Cup campaign has concluded.
"Manchester City and Nottingham Forest have reached an agreement for the transfer of Elliot Anderson," the club said in a statement
"Anderson, 23, is currently competing at the FIFA World Cup with England and has completed a medical in Kansas. The formalities of the move will be finalised upon his return to England.
"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."
The midfielder began his career at Newcastle and had a loan spell with Bristol Rovers before breaking into the first team at St. James' Park.
Financial rules resulted in the Magpies selling Anderson to Forest, where he thrived in his two seasons at the club and grew into one of the best players in the division.
Anderson is currently at the 2026 World Cup with England and has featured in all of the Three Lions' matches so far.