The Portugal international has less than a year to run on his current contract and he is yet to pen an extension.
The 33-year-old has previously stated his desire to agree fresh terms, but Real Madrid have so far not met his demands.
And, according to Calciomercato, Manchester United are now ready to make a move for Pepe this January.
Jose Mourinho knows the centre-back from his time at the Bernabeu and he is aware he could be available at a cut price during the upcoming transfer window.
Premier League rivals Chelsea have also been linked with Pepe, as have sides in Italy and the MLS.