The 26-year-old is currently on international duty with England but limped off injured in Tuesday's 1-1 draw against Germany.
According to reports, City have been keeping tabs on the midfielder for some time and will make their approach when Phillips has returned from international duty.
Pep Guardiola is looking for a replacement for Fernandinho who leaves the club later this month after nine years at the Etihad Stadium.
City have sent scouts to watch Phillips on multiple occasions and believe the Leeds man possesses the attributes to enhance their midfield.
Phillips is contracted at Elland Road until 2024 and last month was quizzed on a possible move away from his boyhood club.
"I've made what I want to do clear. I've made my intentions clear. I want to be at Leeds and I'll focus on that until my contract ends," he told BBC Radio Leeds.
"I'm not really worried about it (transfer speculation) too much. It's always been something I've been confident about sorting whenever it needs sorting.
"There's always going to be noise when you're an England international and you're playing for one of the biggest teams in the country, and I think until the end of the season, I'll be happy at Leeds and always will be."
It remains to be seen whether a City bid would be too difficult for the Leeds-born player to turn down with the guarantee of Champions League football and the opportunity to win trophies.
Manchester United have had a long-standing interest in Phillips, as have West Ham who tried to get a deal done in the January transfer window.
Meanwhile, Leeds have been busy themselves in the transfer market picking up Rasmus Kristensen from Red Bull Salzburg for £8.5million on Wednesday.
This is in addition to the acquisition of Brenden Aaronson who confirmed his arrival at Elland Road last month from the Austrian champions.
Manager Jesse Marsch has moved early in the window to bolster his side after having secured their top-flight status on the final day of the season.