The Colombia international has been struggling for game time at Real Madrid and, having been linked with the Red Devils for some time, may yet end up at Old Trafford.
The LaLiga club is reportedly willing to let the out-of-favor midfielder, who has one year remaining on his current deal, to leave for $29m (£23m/€25m).
The 29-year-old has been the subject of numerous transfer rumors since returning to the Spanish capital following a loan spell with Bayern Munich.
He was linked with a switch to Atletico Madrid last summer, while Napoli and AC Milan were interested, too, but Los Blancos opted to keep him.
Rodriguez discussed his failed move to Atleti in an interview with former England international Rio Ferdinand.
Having only made five LaLiga starts during the 2019-20 season though, he looks certain to leave the club this summer.