It is thought Fischer, 18, has caught the eye of United manager Sir Alex Ferguson and is also being courted by Spanish giants Real Madrid and Atletico Madrid.
But Ajax director of football Marc Overmars wants the Denmark Under 21 international to commit his future to the Amsterdam club.
"We believe in him," former Arsenal winger Overmars told Sporten.dk. "Therefore, we are discussing with Fischer about an extension until 2016 at the moment.
"We want to keep him and we have noticed that he has a big potential."
Fischer scored twice on his full debut to help Ajax win 4-2 at Eredivisie rivals FC Zwolle last weekend.