United are making checks on Camavinga having been alerted to his potential exit from Rennes during this transfer window.
The 18-year-old midfielder has a year left on his contract and if the French side do not agree to a departure, they risk losing a star performer for free next summer.
Ole Gunnar Solskjaer is keen to add a midfielder to his squad but is aware of the club's need to manage their transfer funds carefully during the rest of the window.
Right-backs Kieran Trippier and Max Aarons are also on their radar. According to sources, United will not sign Trippier if his club, Atletico Madrid, demand that his release clause of more than €40m is met.
Sources have told ESPN that United believe they would have to battle Paris Saint-Germain and Real for Camavinga's signature, although his relatively low valuation makes him a more realistic target than West Ham United midfielder Declan Rice this summer.
Rice has so far declined to sign a new contract at West Ham but with a deal until at least 2024, he is still valued at more than £80m.
Sources have told ESPN that Jesse Lingard, who enjoyed a successful spell on loan at West Ham last season, would not be keen to be used as a makeweight in any negotiations to bring Rice to Old Trafford.