The fee is understood to be a Man City club record beyond the £100m paid for Jack Grealish.
Forest have given permission for the midfielder to undergo a medical in due course.
Anderson is currently in the USA with England at the World Cup, where he has started their two group games so far.
Discussions are ongoing about when Anderson can have a medical, given England play on Saturday against Panama in the final game of Group L. The Three Lions are in pole position to qualify as group winners.
Personal terms are not expected to be a problem.
In early June, Man City had an initial offer for Anderson knocked back by Nottingham Forest.
The 23-year-old is currently contracted to the Midlands club until 2029 and has been Man City’s primary midfield target this summer.
Anderson emerged as one of the Premier League’s best central midfielders last season and is part of a group of elite midfield options on the market this summer, including Newcastle’s Sandro Tonali, Crystal Palace’s Adam Wharton and Brighton’s Carlos Baleba.
City admire how Anderson has evolved since joining Forest from Newcastle in 2024, and relations between the two clubs are thought to be excellent.
During England’s World Cup preparation camp, Thomas Tuchel described him as the “full package” and expressed confidence that City’s interest would not distract him.
“He’s a top player,” the England boss said. “There’s nothing more to say, he’s the full package. I’m happy that he’s with us on that kind of level and he’s a key player for us.”
Asked if he was aware of City’s interest, the England boss said: “No comment! He seems not affected. It was an amazing performance [against Costa Rica], so all fine.”
Nottingham Forest are hoping to sign two central midfielders this summer if Anderson leaves.
The plan had always been to add one more option but with Anderson’s move to Man City taking a step forward, it appears two players will come in in his position.
Forest are interested in Tottenham Hotspur midfielder Lucas Bergvall and Inter Milan’s Davide Frattesi, who the Italian club value at around €30m (£25.9m).
They also have a long-standing interest in Celtic’s Arne Engels who, like Frattesi, they tried to sign in January.
