According to the Daily Express, United scouts have been impressed with Rodriguez after taking several trips to Portugal to watch the winger in person.
The 21-year-old has been a target for Sir Alex Ferguson for the past 12 months, and the Old Trafford boss is expected intensify his efforts to sign the Colombian international in the summer.
Rodriguez is believed to have a £36m release clause in his Porto contract, well within United's summer budget, according to the report. Previous reports have also claimed Rodriguez is valued at £36m by his Portuguese employers.
The Porto playmaker has been compared to Cristiano Ronaldo on account of his dribbling, first-touch and devastating ability to make things happen in the final third.
The youngster has impressed in this year's Champions League - Porto face Malaga in the first-leg of their Champions League last-16 tie on Tuesday evening - and has established himself as one of the most exciting, and dangerous, attacking midfielders in the Portuguese top flight.
Manchester United have a strong record when it comes to signing players from the country - Cristiano Ronaldo and Nani were both brought to Old Trafford from Porto's close rivals Sporting Lisbon.
But with Nani set to leave the club at the end of the season, the Daily Express report claims Ferguson could be tempted to include the United winger as part of a swap deal for Rodriguez.
Nani was in great form for United on Monday evening, scoring in his side's 2-1 victory at Old Trafford as United progressed to the FA Cup sixth round.