Mora has made three appearances for his country at this year's World Cup, despite being just 17 years old.
Tuttomercatoweb report that Manchester United are competing with Chelsea, Bayern Munich and Real Madrid for his signature.
Mora will not be permitted to leave Mexico until he turns 18.
Clubs would, however, be able to complete a signing now and loan him back to his current side, Tijuana.
Tijuana are said to want €40 million for the teenager.
United will have another opportunity to watch Mora when Mexico face England in the round of 16 in the early hours of Monday morning.
His performances at the World Cup have accelerated interest from several of Europe's leading clubs this summer.
The competition for his signature reflects the scale of the talent United and their rivals believe they are chasing.
Any deal, though, is likely to remain on hold until Mora reaches his 18th birthday and the terms of a loan-back arrangement with Tijuana can be finalised.