The son of United legend Wayne Rooney, Kai is one of 14 youngsters offered scholarship forms by United, says The Sun.
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Should he commit, Rooney will be on track to turn pro with United when he turns 17 next year.
The contract would see him earn around £50,000-a-year plus an annual signing-on fee. For the moment, the Rooney family are yet to accept the terms.
Kai Rooney plays as an attacker, either as a striker or on a flank.
Also offered scholarship forms is Jacey Carrick, who is the son of United manager Michael Carrick.