The Benfica defender is regarded as one of Europe’s hot prospects.
And Zidane, who has already been busy in the transfer market, is desperate to secure his services.
Eden Hazard has been snapped up along with Luka Jovic but defensive reinforcements are set to be next.
Spanish transfer expert Eduardo Inda says a part exchange deal that includes Real forward Raul de Tomas could be on the cards.
“Man Utd transfer news: Ruben Dias is wanted by Ole Gunnar Solskjaer”
“Real Madrid has asked for Ruben Dias, the Benfica centre-back, and could offer De Tomas,” he told Diario Madrista.
However, Starsport understands United are also ready to cough up for Dias.
Ole Gunnar Solskjaer is tracking Napoli sensation Kalidou Koulibaly but the Serie A outfit value him at £150m.
Dias, in contrast, is available for less than half that amount. Benfica are expected to accept around £60m for their prized asset.
Defence was a major problem area for United last season.
Solskjaer has already signed versatile winger Dan James, but he remains keen on Crystal Palace right-back Aaron Wan-Bissaka and Ajax centre-back Matthijs de Ligt.
Despite United’s overtures, a move for De Ligt looks slim.
Barcelona are throwing their weight behind a deal and mega rich Ligue 1 champions PSG are also jockeying for position.
Signing Dias might appear an easier proposition for United, but De Tomas could hand Real the advantage.
De Tomas isn’t a well known figure outside of La Liga but is rapidly attracting admirers.
Having spent two season on loan at Madrid-based minnows Rayo Vallecano, De Tomas has bagged 38 goals in 66 games.
Benfica are increasingly keen to bring him to Portugal – and any deal that includes the 24-year-old might be given preference by club executives.