According to the Colombian radio station Radio Caracol, the 25-year-old already has an agreement with the Red Devils and his potential move to Old Trafford could be announced after Real's final league match of the campaign against Malaga on 21 May.
They have three matches left and Rodriguez could still play a part in it, especially with Gareth Bale out on the sidelines with a calf injury. Real remain confident that Bale will be fit in time to face Juventus in the Champions League final on 3 June,
This could force the South American star to drop on the bench or be left out of the squad against the Serie A side. Real are determined not to announce his switch to United until the final of Europe's prestigious club competition, reports the Spanish publication AS.
Rodriguez was replaced by Casemiro in the 61st minute in Real's 4-1 victory over Sevilla at Santiago Bernabeu. He waved at all four sides of the stadium while he was walking towards the dugout. He was emotional in Real's final home fixture of the 2016/17 season, fuelling speculation that the attacker could leave his current employers in the summer.
Zinedine Zidane has refused to discuss the suggestion that Rodriguez has already played his final home match for Real. He wants his players to focus on their remaining fixtures as his side are in the race to win the league and the Champions League.
According to Marca, Real are willing to part ways with Rodriguez if the potential suitors could match their €75m (£63.8m, $82.5m) asking price. The same report also claims that newly crowned Premier League champions Chelsea are also looking at the option of signing him.
The Colombian international made a switch to the Spanish capital club after impressing for his national side at the 2014 Fifa World Cup. He has seen his first team appearances limited this season and has started in only 13 league games. Should Real announce Rodriguez's move to United, the former AS Monaco man will be United's first summer signing.