In a further twist to the deal, the Colombia and Monaco star was allegedly spotted boarding a flight from Medellin to Madrid, as pictures emerged on social media.
A picture by Twitter user Juanma Gomez (@jmagomez8) showed Rodriguez in what appears to be an airport.
He said: "I have just seen James on the Madrid to Medellin plane, we are on the same flight".
French newspaper L'Equipe said on Saturday night that Monaco would accept a €90m (£71m) from Los Blancos for the 22-year-old who took the World Cup by storm.
They said a deal could be completed within 48 hours, while Spanish paper AS last week said Rodriguez and the defending European champions had already verbally agreed terms.
World Cup 2014 - Fifa Golden Ball shortlist
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The Spanish paper also reported that Rodriguez is refusing to play in a pre-season friendly for Madrid, fearing a potential injury could ruin his dream move to Los Blancos. Monaco play Junior Barranquillo on Sunday night.
Rodriguez was the tournament's top scorer with six goals - he also added two assists as Colombia reached the quarter-final stage.
He was shortlisted for the World Cup Golden Ball and has also been nominated for Uefa's Best Player in Europe.
Rodriguez cost Monaco €45m (£36m) last summer and he went on to score 10 goals and provide 12 assists in all competitions as Monaco finished second in Ligue 1.
Arsenal manager said Rodriguez was probably too big for Monaco and that a move to Real Madrid or Barcelona was inevitable.
When asked by beIN Sports he said: "I believe that’s true [that Rodriguez is too big for the Ligue 1 club].
"Why? Because a player like that is difficult to keep at a club when you have Real Madrid and Barcelona looking around, and without a doubt there will be people who will be watching them play and thinking about buying him."
Real Madrid completed the signing of another World Cup star, Toni Kroos, on Friday.
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