Rotherham had a bid - reported to be around £800,000 - accepted by the Perth club earlier this week after reviving their interest in the Scotland Under-21 international.
And Saints have confirmed that Wednesday have also been given the right to talk to the forward after Owls manager Stuart Gray revealed on Thursday that his club had submitted a rival bid.
May is now in South Yorkshire to hold contract talks as both Sky Bet Championship clubs compete for his signature.
The 21-year-old looks to have scored his last goal for St Johnstone during their 1-1 Europa League draw against Spartak Trnava in Slovakia on Thursday night, which failed to prevent them crashing out of the competition.
May came through the youth ranks at his local club and scored on his debut as a 16-year-old during a 4-0 First Division win over Airdrie in 2009.
He made his name during loan spells with Alloa - where he netted 19 goals in 22 games - and Hamilton, for whom he struck 26 goals in 35 matches.
The forward returned to Perth to make himself a first-team fixture and netted 27 goals last season before helping St Johnstone win their first major trophy with a Scottish Cup final victory over Dundee United.