The Danish ace is widely regarded as one of the Eredivisie’s top talents, and has been making a major impact at the Amsterdam ArenA in recent times.
This has alerted the Premier League giants, who are both believed to be watching the 19-year-old closely.
Despite his age, Ajax value their teenager at £9m, which has not deterred most of European football’s top sides who are all aware of his potential.
United have been linked with Fischer for some time, with rumours of a bid over the summer having circulated.
David Moyes may look to up his interest in the starlet over the coming weeks, with a need for new bodies at Old Trafford much publicised.
Chelsea are also keen, and could look to launch a move once Kevin De Bruyne’s expected move to Wolfsburg goes through.
Fischer has already netted three goals this term for Ajax in just his second season as a first-team player, and was voted the team’s Talent of the Year at the end of the last campaign.