Rodriguez has not played a league game since October, and his only four appearances since then have all been in cup competitions.
Madrid is keen to reduce its payroll by selling excess players to requirements – and that gives United the opportunity to land Rodriguez at a bargain price, according to The Sun.
The publication indicates that the player was introduced to United, who has long been linked to the attacking midfielder.
But manager Ole Gunnar Solskjaer will have a reduced budget to work with this summer due to the financial impact of the coronavirus pandemic, and must therefore exercise caution in the transfer market.
Borussia Dortmund star Jadon Sancho has been identified as a prime target, but the Bundesliga club will want around £ 80m for the young English winger.
And apart from that, United’s chances of landing at Sancho may depend on their qualification for the Champions League next season.
Rodriguez could therefore be a cheaper alternative – although he is one of the highest earners in Madrid, with a salary of around £ 250,000 a week.
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Everton has also been linked to Rodriguez as the club seeks to sign a big name for manager Carlo Ancelotti.
Arsenal and Juventus have also been linked to Rodriguez, who spent two seasons on loan at Bayern Munich before returning to the Bernabeu last summer.
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