Rodriguez, 24, shares the same agent as newly-appointed United boss Jose Mourinho, Jorge Mendes, meaning the deal could be accelerated now that Colombia have been knocked out of the Copa America.
James Rodriguez and Jose Mourinho share the same agent – Jorge Mendes
Rodriguez joined Real for £56m from Monaco after the 2014 World Cup, but could be available to United for around £50m, according to the Daily Mirror.
Colombia were knocked out of the Copa America on Wednesday, losing 2-0 to Chile in the semi-final in Chicago.
"We look forward to working with him to help nurture that raw talent and fulfil his potential. Eric is at the right club to continue his development."
