Rodriguez signed for Real Madrid from Monaco after the 2014 World Cup.
But despite taking the tournament by storm, he has endured three inconsistent seasons in Spain.
The 25-year-old has struggled for game time under Zinedine Zidane, scoring five goals from 24 games.
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James Rodriguez is wanted by Manchester United
James Rodriguez has scored just five goals so far this season
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Madrid are confident they can recoup the £60million fee they paid to sign him three years ago.
United boss Jose Mourinho feels Rodriguez could help them win the Premier League title next season.
Also keen are European rivals Bayern Munich and Paris Saint-Germain.
James Rodriguez was signed by Carlo Ancelotti back in 2014
Bayern boss Carlo Ancelotti signed the Colombia international for Madrid during his time at the Bernabeu.
He feels a reunion could be just the ticket to help Rodriguez rediscover his best form.
PSG, meanwhile, are also keen and have the financial muscle to broker a deal.