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Phillips, 26, has a contract at Leeds United which runs until 2024 but is reportedly available this summer for a fee of around £60 million.
United are looking for a top central midfielder as they try to relaunch under new manager Erik ten Hag and have placed Phillips as one of their top summer targets.
However, according to The Express, City are also interested in Phillips and could pip the Red Devils to his signature.
The report goes on to say that Pep Guardiola sees the Englishman as the perfect replacement for Fernandinho who is set to head back to Brazil at the end of the season.
Phillips may choose the Etihad over Old Trafford with City in much better health compared to United.
While City have won three Premier League titles in four years, United are likely to finish sixth this season and are five years without a trophy.
Meanwhile, it is believed Phillips is reluctant to join the red side of Manchester because of their long-running rivalry with his hometown club Leeds.
Losing out on another transfer target would be a bitter blow for United, who are already being priced out of a deal for West Ham’s Declan Rice.
The Manchester giants were hoping to sign Rice this summer, but Hammers manager David Moyes has suggested he wants £150m for his captain.
With reports stating Ten Hag will have a reduced transfer budget to work with, it remains to be seen whether the Dutchman is prepared to spend such a significant amount on one player.
It is thought his other targets include a striker, centre-back and two back-up goalkeepers.
To help fund a squad overhaul, Ten Hag may look to move on the likes of Aaron Wan-Bissaka who has fallen out-of-favour this season.
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