The Premier League side sent scouts to watch the Portuguese club beat Academica 2-1 on Sunday night, with Rodriguez getting on the score-sheet.
The 21-year-old had an impressive game, which could prompt United to make a move in January.
Sir Alex Ferguson is believed to be an admirer of the wide-man and sees him as the ideal replacement for Nani, who is likely to leave the club following a series of below-par performances.
Porto are keen to keep hold of their star player, who has a buy-out clause in his contract believed to be between £25m and £38m.
This has not deterred the Scot, who could still press ahead with a winter approach, but he is unlikely to match the figure quoted.
Fergie appears convinced that the South American, who arrived at Porto from Argentine club Banfield in 2010, is the ideal man to provide further ammunition for Wayne Rooney and Robin van Persie.
Since his arrival in Europe, Rodriguez has averaged nearly a goal every other game, operating from a wide position.
Although he is primarily a winger, he is able to play as a central attacking midfielder or a in a striking role.