May, a £400,000 signing for the Dons from Preston in the summer of 2017, looks to have slipped down the pecking order after Aberdeen manager Derek McInnes secured the signing of Curtis Main and the return of James Wilson on a permanent deal.
The Scottish Sun is reporting that Dundee, Dundee United, St Johnstone and St Mirren are all keen on the striker.
The Tannadice club would likely have to offload a player in order to make room for May in terms of squad size and wages, with Osman Sow likely to get the bullet after a disappointing six months.
Dens Park boss James McPake is keen to sign a replacement for the departed Kenny Miller, but the two Saints will also be keen on acquiring the striker's services.
The Dons will listen to offers for the Perth-born forward, who has two years left on his Pittodrie contract.