The 23-year-old is currently on duty with the Three Lions at the World Cup.
He played the full 90 minutes of the 4-2 win over Croatia and 73 minutes of the goalless draw with Ghana before being withdrawn for Eberechi Eze.
After being granted permission by Forest, Anderson will undergo a medical in New York on Friday, with personal terms yet to be agreed.
Forest had rejected two previous bids for the player but accepted a deal that could ultimately be worth up to £130m, which would represent a British record fee.
Anderson is expected to start in England's final group game against Panama on Saturday.
Speaking to BBC Sport, the former Newcastle United man explained he is fully focused on helping England progress at the World Cup.
"It's pretty easy really, I'm just focusing on the present, I'm blocking it all out," he said.
"I've got a plan and it's to perform for England. I'm putting myself in the best position to cross the line and do that."
Anderson made 88 appearances for Forest, scoring four goals and bagging four assists.
A Fabrizio Romano post read: "Manchester City agree deal to sign Elliot Anderson for a transfer fee worth £116M!
"Deal at final stages for days as revealed last week and done right now between clubs for total potential fee up to £116m.
"Anderson asked Nottingham Forest today to leave the club, wanted Manchester City move and medical will be in the US."
Jordan Henderson has called on England to remain focused and disciplined as they enter the knockout stages of the 2026 FIFA World Cup, warning his teammates that there will be no easy matches on the road to glory.
Felix Nmecha has become one of the most closely monitored midfielders this transfer window, with elite clubs across Europe weighing up a potential move for the Borussia Dortmund player.
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Springbok head coach Rassie Erasmus has tackled England as both a player and a coach and says it is always something special when the two nations do battle.
