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He played the full 90 minutes of the 4-2 win over Croatia and 73 minutes of the goalless draw with Ghana before being withdrawn for Eberechi Eze.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."
New Zealand took firm control of the series-deciding third Test against England after a superb opening day at Trent Bridge, finishing on 361/4 after choosing to bat first.
New Zealand made three changes to their starting XI for the third Test match against England at Trent Bridge on Thursday.
Springbok coach Rassie Erasmus and captain Siya Kolisi are under no illusion as to the challenge that awaits when England arrive on the shores of the Republic next week.
Ghana put on a defensive masterclass as they held star-studded England to a 0-0 draw in their Group L match at Boston Stadium in Massachusetts on Tuesday.
