The 21-year-old is being pursued already by Rotherham United, with the Sky Bet Championship team understood to have made an offer of £1.2m last week.
Talks have been ongoing between Rotherham and St Johnstone but the dialogue may now have been complicated by the rival offer made by Wednesday. That might also have been considered an unwelcome distraction for the Perth club as they prepare for the second leg of their Europa League third-round qualifier against Spartak Trnava tonight. The Slovakian side are at home and lead the tie 2-1.
Wright, however, intends to field May in the match - the striker missed the first leg due to a thigh injury - and is hopeful that his young player will be inspired instead to lead his side into the next phase of the competition. May scored 27 goals last season and a couple more tonight could be enough to take St Johnstone into the play-off stage.
"Sheffield Wednesday have officially contacted the club, The board will deal with that," said Wright. "I've been kept up to date but I did say to the chairman that if any deal was to be done this week we have to make sure we have Stevie available for Thursday.
"He is important to us. He gives us a spark and he scores goals, which we need. The most important thing for me, is that he wants to play in the game. If it turns out that this is his last game, let's hope he can make the right headlines."
St Johnstone have offered the Scotland under-21 internationalist a new contract and May intimated last term that he was willing to sign an extension to his deal. It seemed unlikely that another club would not make an offer and disrupt those plans, though, and the initial offer from Rotherham suggested that May would not begin the league campaign in Perth.
"When a player is in the last 10 months of his contract and there are two clubs in for him, it's going to be difficult for the club to turn it down," acknowledged Wright. "Stevie is focusing on the game. Whatever happens between any club and ourselves regarding Stevie will go on in the background.
"But at the moment Stevie's on the plane [to Slovakia] and he will play. When he gets back something may or may not have happened."
He also may or may not still be involved in European competition. It is uncertain what the impact of having another Europa League tie to play this season might have on the striker's ambitions, but there are those in the St Johnstone squad who hope to ensure that their coveted colleague has a difficult decision to make on his future.
Steven MacLean, for now May's strike partner, is one of them. "I'm hoping Thursday won't be Stevie's last game and that he will stay," said the 31-year-old, who spent three years at Wednesday earlier in his career, winning the League 1 play-off final in 2005.
"Maybe it could be a factor in our favour if we get through this one and get one step closer to the group stages. Perhaps the lure of those big nights might persuade him to stay for a bit longer."
English football continues to represent the logical next step in the career of a young Scottish player, although MacLean is adamant that May would be better served by remaining with the McDiarmid Park side for a little longer, should they extend their Europa League campaign. "The Europa League would be a great stage for him," added MacLean.
"Everyone wants to play at the highest level they can. If we did get to next round and, if we were lucky, then the group stages it would definitely cross his mind."
To get there St Johnstone must first cross the white line in Slovakia and score at least two goals. It is perhaps a daunting prospect for a club which found Spartak to be troublesome opposition last week, but MacLean is optimistic that his side can still progress.
After all, this St Johnstone team have produced unexpected results in Europe before. "We've been to Rosenborg, to Minsk and to Lucerne and got good results," said the striker.
"The difference was in those game we were away from home first. But we know we've got the team to score goals."