The attacking midfielder was one of the stars of the show in South America as he fired his side to the quarter-finals.
Although Rodriguez netted six goals in five games, it wasn’t enough to avoid elimination at the hands of hosts Brazil.
But the 22-year-old is still likely to be in the running to be named as the tournament’s top performer, and United are ready to swoop for his services.
The classy starlet only joined current club Monaco last year in a big money £45m switch, and has already indicated that he would prefer a move to Spanish football rather than the Premier League.
Yet this has not deterred Louis van Gaal, who is keen to make the Colombian a key member of his new look Red Devils side.
United are prepared to offer £65m to Monaco, which could be sweetened by add-ons which may see the fee rise to around £90m in the future.
The Ligue 1 heavyweights are heavily backed by their Russian owner, but the lure of such sums may be tempting for them with Financial Fair Play a worry.