City have had their bid of £42million, with a possible extra £3million in add-ons, accepted by the West Yorkshire outfit.
The two clubs have also come to an agreement for City academy midfielder Darko Gyabi who is set to move in the opposite direction. Both deals are now subject to personal terms being agreed.
Once both deals are signed, sealed and delivered, Phillips will become City's third major signing of the summer.
Pep Guardiola's side have already secured the signatures of strikers Erling Haaland and Julian Alvarez as they prepare for the forthcoming season.
City have been on the lookout for a midfielder following the departure of Fernandinho who ended his nine-year stint at the Etihad last season.
Phillips made his name at Leeds after progressing through the club's youth ranks. The 26-year-old played a crucial role in United's promotion to the Premier League two years ago.
After establishing himself in the top tier, Phillips has gone on to feature regularly for England under manager Gareth Southgate.
Over the last two seasons with the Whites, the midfielder has 49 games in the Premier League, scoring one goal and producing three assists. The player averages 48 passes per match and has also made 394 recoveries.
Since breaking into the Leeds first-team, Phillips has made 234 appearances for the club across all competitions, scoring 14 goals.