The 19-year-old star had trials for Chelsea when he was 12, but was spotted by a River Plate scout in England and signed for them instead.
United are said to have sent scouts to watch Mori in Argentina, while City have lodged their interest with River Plate.
As usual, Real Madrid, AC Milan and Barcelona have also been mentioned among the interested parties.
Mori has scored nine goals in 31 appearances for River Plate.
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Meanwhile, Talksport reports Tottenham and Seville are in the mix to sign Getafe's Derek Boateng.
The Ghana midfielder is wanted by Spurs and Seville, but they'll have to wait until next summer to fight it out.
Getafe's president Angel Torres has slapped a £5 million pricetag on Boateng, and Spurs are ready to get the White Hart Lane credit card out.