According to the Italian publication Mediaset Premium, the Colombian international has already agreed personal terms with the 20-time English champions over a proposed move to Old Trafford in the summer transfer window.
The Red Devils are yet to agree a fee with the Champions League and the La Liga winners for the sale of Rodriguez. However, it is reported that the two clubs are "not far away" from reaching an agreement for the transfer of the former FC Porto and Monaco star.
Rodriguez saw his first team appearances being limited under Zinedine Zidane last season. He started in 13 league games and has made 33 appearances across all competitions for the Spanish capital club. He scored 11 goals and registered 12 assists in the process.
AC Milan were interested in signing the 25-year-old playmaker, according to the Spanish publication Mundo Deportivo. Real are ready to part ways with Rodriguez if the potential suitors can meet their asking price of €75m (£68.5m, $75.4m).
The Serie A outfit are not willing to meet the 12-time European champions' asking price. Milan have proposed an alternative, which will allow them to sign Rodriguez on a season-long loan and then make his move permanent a year later. They want the asking price to be reduced, which is something Real may not accept.
Milan's struggle to land Rodriguez has seen the player set his sights on joining Jose Mourinho's side. The Portuguese tactician also wants to land Morata to strengthen United's attack for the next season.
The former Inter Milan and Chelsea manager gave Morata his Real first team debut when Mourinho was in charge at Santiago Bernabeu. The United boss now wants to reunite with the Spain international at Old Trafford. Should United manage to land Morata and Rodriguez this summer, the addition of the Real star duo will bolster the Red Devils' attack for the next season.