Manchester United are reportedly preparing an opening bid of around £43m for the 23-year-old France international, who has fallen down the pecking order in the Spanish capital after Mourinho's arrival and was left out of France's World Cup squad this summer.
The move would represent a swift reaction from Michael Carrick's side, who have already strengthened their midfield this window with the additions of Youri Tielemans, Andrey Santos and, most recently, long-term target Carlos Baleba from Brighton.
The timing points to a response born of necessity rather than long-term planning. United endured an underwhelming performance against Hull, with both Tielemans and Santos substituted together, prompting suggestions that further reinforcements are needed in the middle of the park even after a busy summer of recruitment.
Camavinga, a two-time Champions League winner, has spent five years at the Bernabéu and can also operate at left-back, giving United's move added versatility even if a deal for a recognised central midfielder remains the priority.
Real Madrid have shown some willingness to consider offers as they reshape the squad under Mourinho, though Camavinga himself has previously expressed a preference to fight for his place in the Spanish capital rather than seek a fresh start elsewhere, telling those close to the situation that he wants the chance to win over his new manager before considering any exit.
Reports in Spain in recent weeks have suggested Real would be open to a fee in the region of £42m for Camavinga, broadly in line with the opening bid United are said to be preparing, though any deal would still require the player himself to sanction a move he has so far resisted.
United's interest also comes against the backdrop of the club's continued pursuit of a long-term defensive midfield option, with Camavinga's versatility across midfield and left-back offering a squad solution that a more specialist target might not.