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United's record goalscorer posted a picture on social media of the 11-year-old, the oldest of Rooney's four sons, signing a contract at his father's former club.
Rooney accompanied the image with the caption: "Proud day. Kai signing for @manchesterunited. Keep up the hard work son."
Rooney, 38, now in interim charge of Championship club Derby, joined United from Everton in 2004 as a teenager.
The former England striker had a glittering 13-year career at Old Trafford, scoring 253 times in 559 appearances and breaking Bobby Charlton's long-standing goals record in January 2017.
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Coleen Rooney, Wayne Rooney's wife, wrote on Instagram: "Special night..... congratulations Kai. I love you and I am so proud of you. Keep trying your best".
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