The 21-year-old moved to Rennes in 2015 but has struggled to establish himself in the first team and may be eyeing a move elsewhere to gain more game time.
A number of clubs have reportedly been in touch with the Ligue 1 outfit about signing the right-back this month, with Ligue 2 title challengers Reims said to be keen.
However, according to L’Equipe, Cavare is also attracting interest from Rovers, who are desperate to shore up a leaky defence that has seen them slip to 22nd in the table and in danger of relegation.
The Lancashire outfit have reportedly opened talks with Les Rouges et Noirs and negotiations are set to take place over the coming days.
Burnley midfielder Jack Cork is confident his side can reach the Europa League but expects a stern test in qualifying.