Pepe, who is in his eighth season at the Bernabeu, has been a regular under Carlo Ancelotti, but Varane's emergence could leave him warming the bench in the future.
Chelsea have also been keeping a tab on Pepe, as they seek to strengthen their defence.
But according to Marca, United are set to miss out on Pepe who is ready to commit his future to Los Blancos.
The 31-year-old, who is out of contract next summer, is set to sign a new deal that will keep him at the club until 2017 albeit he wants to stay until 2018.
GETTY
Pepe is set to stay at Real Madrid
Though a prospective new deal for Pepe would represent a blow for United, they won't give up on Varane.
Recent reports have suggested that Varane, who is valued at £30m, is keen to move to Old Trafford.
Paris Saint-Germain are also monitoring his development.
We use your sign-up to provide content in ways you've consented to and to improve our understanding of you. This may include adverts from us and 3rd parties based on our understanding. You can unsubscribe at any time. Read our Privacy Policy
