Burke twice put Forest ahead against Wigan before Thomas Lam sealed Saturday's 4-3 win in stoppage time.
Sunderland are reportedly interested in the 19-year-old, who earned his first senior Scotland cap after making just 19 first-team appearances for Forest.
"His future is at Nottingham Forest," Montanier told BBC Radio Nottingham.
"Oliver knows that the most important thing for young players like him is to play and to progress. Here at Nottingham Forest, it is a good place to play and improve before going away.
"A lot of young players go away very early and then they have problems trying to improve to play at the top level.
"Oliver knows that it's important for him and for us that here we have good facilities and a good atmosphere around him to improve his level and learn the game."
Burke, who came through Forest's academy, signed a four-and-a-half-year deal with Championship club Forest in February.
Teenage winger Oliver Burke will improve his career prospects if he remains with Nottingham Forest, insists Reds boss Philippe Montanier.