The Wolfsburg defender was one of the stars of his nation’s World Cup showing, with the statistics stating that he put in one of the performances of the tournament against Argentina.
United appear to have been alerted by this, and have put together a ‘concrete’ offer for the 21-year-old.
Any swoop would come as a major surprise, with the Red Devils having already moved to snap up Luke Shaw this summer.
The teenage England international cost £30m, and is likely to share the left-back berth with Patrice Evra next season.
Rodriguez is seen as one of Europe’s most promising full-backs, and has also been linked with Liverpool and Chelsea, as well as a host of other top sides.
The young defender joined Wolfsburg in 2012 from FC Zurich, and has been one of the Bundesliga’s top stars in his role since then.