The Red Devils are interested in signing a centre-back in the summer transfer window to bolster their backline.
The ongoing situation with Eric Bailly continues to rumble on with the Ivorian likely to leave after falling out of favour this season.
But the club are happy with the progress Victor Lindelof has made and want to partner him with a more experienced centre-back.
There are a number of names that United have pinpointed as potential signings including Kalidou Koulibaly, Toby Alderweireld and Varane.
The Frenchman, though younger at 25, has considerable experience at the most elite level.
Varane has had an incredibly successful past few years, winning the Champions League four times in five seasons and helping France to a World Cup in the summer.
But his future is being discussed at length in recent weeks after reports indicated that he was unhappy at the Bernabeu.
Zinedine Zidane will bring in Eder Militao this summer – a move that could marginalise the former Lens man.
But Varane has insisted that this is not the case.
“I’m at Real Madrid. The biggest club in the world,” he said.
“I’m constantly in a state of challenging myself because the requirement here is that of the highest level.
“I’m focused on my career. We had a tough time in the middle of the season where we had to fight.
“I’ve given my all in this shirt and I can be proud of that.
“I cannot control everything that’s said [in the media]. Many things aren’t right but I can’t say I’m unhappy in Madrid.
“We had a tough season, it’s the daily routine for players at the top.
“I’m discreet and the problem when you’re a quiet person is you hear many things but talk little about yourself.”