The Sun says United are planning a summer move for the England midfielder.
A source said: “United are watching the situation with Phillips very carefully.
"He is likely to go out on loan in January, but he will then be looking for a permanent switch in the summer, and Ten Hag wants to be right at the front of the queue.
“He is a fan of the way Phillips plays, and thinks his style is perfectly suited to United and to the philosophy he is trying to instil in the group.
“Plus, the new investors want the club to focus on investing in and developing English players, and Phillips still remains one of the best English players in the Premier League.
"It wouldn't be the first time a player has moved from City to United, but it's always controversial."
The source added: “There's a feeling that Guardiola wouldn't stand in Phillips' way.
“The move to City simply hasn't worked out and Guardiola is keen for the situation to be resolved as fast as it can be.
“Midfield is an area that United still need to improve on. Sofyan Amrabat has done OK since coming in on loan but he's not seen as a long-term United player.
"Phillips would slot in that role if United can bring him in.
"There's also a question mark over Casemiro. He's tied to a long term deal, but if an offer came in for him United might be willing to listen because Casemiro is on massive wages."