Mundo Deportivo says Real have outbid Manchester City as they close on a deal on the 20 year-old.
City had offered €2m for Soro, but Real have since bettered that by €500,000.
And Soro is expected to commit to Madrid, where he will join Raul's Castilla team for next season.
Soro played 29 games in the Segunda Division last season with Zaragoza.
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