The 17-year-old star has impressed for Brazilian side Palmeiras in recent months.
Souza has also been shining on the international stage with Brazil’s Under-17 side.
Spanish newspaper AS last week claimed Real Madrid have agreed a deal to bring the attacking midfielder to the Bernabeu next summer.
Souza will reportedly go straight into Real’s B team before eventually linking up with the seniors in Madrid.
However, according to Spanish transfer outlet Don Balon, Jose Mourinho has other ideas.
They say the United boss is desperate to beat Real to the Brazil ace to bolster the Red Devils’ youth academy.
The playmaker, who is under contract until 2019 at Palmeiras, has a £45m release clause in his current deal.
Real president Florentino Perez is trying to negotiate a lower fee for Souza, who will enter the final year of his contract next season.
United were back in action last night as Mourinho’s men beat Swansea in the fourth round of the Carabao Cup.
The Red Devils host Tottenham in the Premier League on Saturday, with United looking to mount the pressure on leaders Manchester City.
