The Daily Mirror reports that Reds boss Kenny Dalglish faces competition from the likes of Real Madrid, AC Milan and United in the race for the 19-year-old.
Liverpool scouts have watched the midfielder in action five times in recent months.
Liverpool director of football Damien Comolli personally watched the youngster make his Brazil debut against Scotland.
Rodrigues’ current contract with Brazilian side Sao Paolo is set to expire in the summer of 2016.
