The Premier League champions are understood to have agreed a deal to sign the England international for an initial £42m, which could rise to £45m with performance-related add-ons.
Such a move would bring Phillips's 12-year stay at Elland Road to an end, during which he played a key role in helping the Whites return to the Premier League.
The 26-year-old has become a mainstay of the England setup since his Premier League breakthrough, but he has been strongly linked with an exit from Jesse Marsch's side after the Whites secured their top-flight survival on the final day.
According to the Daily Mail, Phillips is poised to complete his Man City medical on Friday, with the move being delayed while the midfielder has been enjoying his holidays.
Kalvin Phillips 'to complete Man City move this week'
Phillips's Elland Road contract is due to expire in the summer of 2024, and the report adds that the Englishman would have been prepared to commit to fresh terms, but talks were not opened.
West Ham United, Aston Villa and Newcastle United were all said to have been interested in a move for Phillips, but the 26-year-old was not interested in a move to any of those teams.
Phillips prepares to depart Elland Road after scoring 13 goals and setting up 13 more in 233 appearances for the club in all competitions, although he struggled with injury problems in the 2021-22 season.
The former Wortley Juniors youngster still managed 20 top-flight appearances - 18 of which came from the first whistle - and be played the full 90 in each of the Whites' last six Premier League matches.
Phillips will arrive at the Etihad Stadium as a replacement for Fernandinho, who has departed as a free agent to rejoin Atletico Paranaense 17 years after leaving the Brazilian side for Shakhtar Donetsk.
Pep Guardiola's side have only welcomed one new player to the club so far this summer in Erling Braut Haaland, but more could follow Phillips through the door in the coming weeks.
The left-back area has proved particularly problematic for Guardiola over the past couple of years, and Brighton & Hove Albion's Marc Cucurella is thought to be at the top of their list for defensive reinforcements.