Mastantuono, 16, has already scored two goals in eleven competitive matches for River at senior level.
Sport says Real Madrid who are leading the hunt for Mastantuono.
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Although Real Madrid have yet to make a bid, chief scout Juni Calafat and director Jos Angel Sanchez have contacted River Plate to discuss a deal.
The 16-year-old has a 45m buyout clause, but that rises to 50m in the final stages of the summer window.
In addition to Barcelona, Manchester United, Manchester City and Chelsea are also keen on Mastantuono.