The 20-year-old, who won his first senior caps for Uruguay in November, made his first division debut in 2007 and helped Nacional win the 2008/09 league title and this season's Apertura tournament.
After captaining his country at the World Youth Cup in Egypt last year, Lodeiro made his senior debut in the 2010 World Cup qualifying playoff they won 2-1 on aggregate against Costa Rica.
Lodeiro will link up with Uruguay striker Luis Suarez at the Dutch club.
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