The 18-year-old forward is one of the world's hottest properties, and only recently signed a new Ajax deal in February. But that hasn't put off potential suitors, with United and Bayern set to go head-to-head this summer.
The report claims Bayern Munich could kick off proceedings with a summer bid for Fischer - whose contract expires in 2017 - which may prompt a response from United.
Old Trafford scouts have been closely tracking Fischer, one of the brightest prospects to graduate from the famous Ajax academy in recent years.
The teenage forward already has two Denmark caps, and along with fellow star and international teammate Christian Eriksen, has fired Ajax to the top of the Eredivisie.
Fischer has made 20 appearances in the league so far this term, scoring eight goals - Ajax's third highest scorer behind Eriksen and Siem de Jong.
Equally comfortable through the middle or on the left, he's enjoyed a break-out campaign so far.
Premier League scouts flocking to Amsterdam to watch Eriksen have left just as impressed with Fischer, and the Dutch giants may face a battle to keep hold of him this summer.
Pep Guardiola is set to take the reigns at the Allianz Arena next season, and with his record of bringing through youth talent, a move for Fischer could be on the cards.