The 18-year-old has made 10 appearances for the Ligue 1 club this season and French media believe he has caught the Red Devils' attention.
Le10Sport reports that the Old Trafford club have expressed interest in the centre-back and are moving to the front of the queue of suitors, with several big clubs across Europe said to be keen.
However, the outlet says that United will need to present a bid of €30 million (around £25 million).
Additionally, Saint-Etienne would demand that Saliba remains at the Stade Geoffrey-Guichard on loan for next season once a deal has been agreed.
The France Under-19 international has broken into the first team this season, making 13 appearances in all competitions since his debut in September.
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