The 23-year-old Switzerland international has emerged as one of the top players in his position over the course of the past few years since joining the Bundesliga club from FC Zurich in 2011.
And reports claim that United are eager to offer him a fresh challenge, with Luke Shaw’s injury woes having limited Louis van Gaal’s options at left-back.
The English youngster suffered a horror leg break in Champions League action against PSV Eindhoven earlier this season and is unlikely to return before the campaign ends.
Rodriguez is seen as a good option to slot into the role, although it’s not known if LVG sees him as a long-term first choice player.
The defender – who is known for his set-piece prowess – was part of the side that knocked United out of the Champions League before Christmas.