The 18-year-old, who made his UEFA Champions League debut in the 2-2 draw against Manchester City at the Etihad Stadium earlier this season, is currently in talks with the club's hierarchy.
Head coach Frank de Boer is keen to tie down the Denmark international to a new long-term deal to ward off potential suitors.
Fischer has been reportedly scouted by the likes of Manchester United, Chelsea and Inter Milan while he was playing for the Denmark Under-17 side.
But, after setting the Eredivisie alight with an impressive eight goals in 18 games this season, the talented player looks set to stay in Amsterdam.
He told Danish website Sporten.dk: "The negotiations are going quite smoothly and I do expect everything to be sorted out soon.
"I don't know when it will happen, but I'm sure it will be over very soon."