Colombia international Rodriguez has been strongly linked with a move to Old Trafford after falling out of favour under Zinedine Zidane.
According to Bleacher Report, Mourinho will discuss the transfer with agent Jorge Mendes.
Mendes, who represents both Mourinho and Rodriguez, is confident a deal can be struck.
The 25-year-old is reportedly seen as a better fit than Juan Mata for the United team next term.
Real signed Rodriguez from Monaco for £63m in 2014 following his exploits at the 2014 World Cup.
But he has gradually seen his first-team action reduced each season, managing just 33 appearances in all competitions this term.
Despite his limited game time, Rodriguez has still managed 11 goals and 13 assists.
He is under contract until 2020 and recent reports in Spain suggest he has told pals he is leaving Real.
Earlier this month, Rodriguez's stepfather backed him to leave Real.
“I’m sure James knows about how few minutes he's been getting,” Juan Carlos Restrep said.
“I, as a father, prefer him to be a starting player in a different team than being a substitute at Real Madrid.”
Marca report today that Mourinho wants to swap Alvaro Morata in exchange for David De Gea.
Real are hoping to sign De Gea after seeing a move fall through in 2015.
