Manchester City have already had two bids rejected and are preparing to return with a revised offer. Their most recent proposal stood at £110m guaranteed plus add-ons of £11m — a figure Forest's owner Evangelos Marinakis rejected, insisting on a British record total package.
Fabrizio Romano confirmed earlier this week that the deal is now at its "final stages," with City confident an agreement will be reached.
A completed deal at the figures being discussed would surpass the £125m Alexander Isak commanded in his move from Newcastle United to Liverpool last summer, and would sit above the £105m Arsenal paid for Declan Rice, the current record paid by a British club.
Anderson, 23, joined Nottingham Forest from Newcastle for a net £15m two years ago after Newcastle were forced to sell due to Premier League financial regulations. Since moving to the City Ground he has developed into one of the best midfielders in the country, earning a regular place in England's starting line-up.
He is understood to have agreed personal terms with City on a contract running to 2031. England manager Thomas Tuchel has confirmed he has no objection to players attending medicals during the World Cup, removing a potential logistical obstacle.
City see Anderson as the cornerstone of incoming manager Enzo Maresca's midfield rebuild. With Bernardo Silva having left on a free transfer and Rodri entering the final year of his deal, the club are determined to secure a player around whom the post-Guardiola era can be constructed.
Manchester United had also been interested in Anderson but appear to have stepped back, leaving City as the clear frontrunners.