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Four clubs - believed to be Southend United and Brentford plus two non-league teams - have expressed an interested in signing the 22-year-old and could possibly nab the player for next to nothing.
The 22-year-old's contract expires in the summer, and although contract negotiations have been on-going since last December, it is believed May and his agent rejected The Lions' initial offer.
Millwall director Constantine Gonticas hopes a deal can be sorted out soon, adding: "His attitude has been spot-on. He is a developing player and we have been extremely impressed with his progress.
"His goal in the 2-1 win over Luton showed what he is capable of and he is very much the kind of player Millwall need for the future."