Despite the playmaker being available for transfer, the Colombia international had been due to feature in Madrid's squad for the game with Athletic Bilbao on Sunday afternoon.
However, Rodriguez was left out of the travelling party, with Zinedine Zidane later revealing that the 28-year-old had asked not to be involved.
According to the Daily Mail, United are one of a number of teams who would have taken notice of the development.
Although the Premier League giants are well stocked for options in the final third, Ole Gunnar Solskjaer is expected to bring in at least one high-profile name during the next window.
Los Blancos are said to value Rodriguez in the region of £22m.