United are looking to bring in the Colombia international as a potential replacement for Portuguese winger Nani.
Nani has been linked with the likes of Inter Milan and Juventus in recent months and could decide to leave United this summer.
Rodriguez has admitted he would be interested in a move to Old Trafford and United scouts watched the 20-year-old in action several times last season.
United face competition from Bayern Munich in the race to sign the highly-rated midfielder.
