Spanish newspaper AS claim that Wayne Rooney's move to Everton has allowed United to intensify their interest in the Colombian.
Rodriguez has been linked with United for the past 18 months after struggling to fully establish himself at the Bernabeu since his arrival in 2014.
The 25-year-old wants his future sorted ideally before the start of Madrid's pre-season campaign this week and may get his wish from United, who have already splashed out £75m to land Romelu Lukaku.
But they will have to shell out another £62m if they wish to prise Rodriguez away from Spain, according to AS.