The Daily Star Sunday says there is concern at The Etihad Stadium about the number of home-grown players there will be in their squad for the start of next season.
Premier League squads must include eight in a 25-man pool.
Barkley, 20, has been on City’s radar for some time, but he is also interesting his former Everton manager David Moyes at United.
Indeed, ex-United academy chief Eric Harrison declared last week that Moyes had to make winning the race for Barkley a priority.
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