According to the BBC, the 23-year-old England international is expected to undergo a medical before completing his move to the Etihad Stadium.
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Forest accepted City's third offer after rejecting two previous bids, with the transfer set to surpass the £125 million Liverpool paid Newcastle United for Alexander Isak.
Anderson joined Forest from Newcastle in 2024 for £35 million and made 88 appearances, scoring four goals.
He played a key role in helping the club secure a seventh-place Premier League finish in 2024/25 and reach the Europa League semi-finals the following season.
Forest are now expected to reinvest the transfer fee, with Tottenham's Lucas Bergvall and Inter Milan's Davide Frattesi among their midfield targets.