The Colombian has been widely rumoured to be unsettled at the Bernabeu, and United have reportedly had discussions with Rodriguez's agent, Jorge Mendes, over taking him to Old Trafford, per John Richardson in the Express.
The report indicated that Los Blancos boss Zinedine Zidane would be open to Rodriguez leaving the club, but Real will demand a similar fee to the £63 million they paid Monaco for him in 2014.
Bleacher Report's Dean Jones recently rubbished the idea that Rodriguez could move to United—not least because Real's transfer ban comes into play in the summer:
However, the rumours have yet to die down about a possible move.
Certainly the 24-year-old is not enjoying the happiest of seasons in Madrid having fallen out with Zidane's predecessor, Rafael Benitez, per Richardson.
He also does not seem to have enamoured himself to the Frenchman since his instalment in the top job and has played only 54 minutes under the new boss in his first two matches in charge, per WhoScored.com.
Rodriguez would undoubtedly be guaranteed more game time if he made the switch to Old Trafford, as the Red Devils are currently lacking flair and invention in attack.
Rodriguez's addition could go a long way to rectifying that as he is a very skilful attacking player and can operate either on the flanks or in the No. 10 role.
As both a goalscorer and creator he would be a terrific asset at Old Trafford. His 13 goals and 13 assists in La Liga last season a testament to how effective he can be in front of goal, per WhoScored.com.
United have been dour to watch under Louis van Gaal this season, and their goal return at home is startlingly poor, per Squawka:
Squawka Football @Squawka
Man United have now failed to score in six of their last 11 games at Old Trafford across all competitions. https://t.co/YYQi9XiiFD2016-1-23 23:20:03
Rodriguez would certainly bring more creativity—which United fans would hope equates to more goals—but it is unclear whether he would be allowed the freedom he desires under Van Gaal.
A Rodriguez switch to United would be a major move, and it remains an unlikely prospect, but if he is genuinely not enjoying himself at the Bernabeu he could look to force a move away.
Per Gianluca Di Marzio (per Will Giles in Metro), United's Italian defender, Matteo Darmian, has discussed a move to United with his compatriot, and Perin could switch to Old Trafford as David de Gea's successor.
There is still a possibility that United's Spanish No. 1 could move to Real in the summer if their transfer ban is lifted or delayed, per Sport Witness.
As a result, United have continued to look out for his potential replacement, and the 23-year-old Perin is being viewed as an option.