Any move has the approval of Jose Mourinho, who has not been fully convinced by Camavinga's performances in recent seasons.
Manchester City have received Real Madrid's proposal and are weighing up whether to move for the midfielder, who is well regarded in the Premier League for his range across the pitch and physical power.
City have already strengthened that area of the pitch this window, paying €135 million for England international Elliot Anderson, a huge fee for a deep-lying midfielder. The Etihad club were also interested in Italy's Sandro Tonali, but it is Tottenham who look set to land him, in a deal worth more than €100 million to Newcastle.
Real Madrid see a possible Camavinga departure as a useful opportunity in this market, with City still hunting for midfield reinforcements and Premier League clubs able to pay well.
Mourinho has given the move his blessing and has no objection to Camavinga leaving, with the funds potentially used to bring in another midfielder at the Bernabeu. The Portuguese coach still believes his Real Madrid squad needs two further additions, one in midfield and another at centre-back.
Manchester City have also spent weeks working on a move for Ayyoub Bouaddi, the Moroccan prospect at Lille. Contact between the clubs are already under way, with City considering whether to sign him immediately or leave him on loan at Lille for another season.
Real Madrid have so far allowed three players to leave compared with last season.
Dani Carvajal and David Alaba departed when their contracts expired, while the club granted Dani Ceballos a free transfer, leaving him able to negotiate his next move.
Further departures remain possible, with Fran Garcia and Raul Asencio the most likely names to follow.
Camavinga has been a regular fixture in Madrid's squad, but Mourinho's willingness to sanction his sale points to a shift in how the head coach views his midfield options heading into next season.