Aberdeen, Dundee and Hearts have all intimated their desire to land the Scotland international, either during the current transfer window or in the summer.
But May insists that Hibs are the only Scottish team who could come up with the funds to convince the defender to sign before the February 1 deadline.
May firmly believes, however, that Barr will remain at Westfield for the rest of the campaign and will not allow his focus to be diverted from the task in hand.
Speaking ahead of Sunday's meeting with Dundee United, May said: "I don't think Darren Barr is a difficult situation, because somebody will need to go and pay money.
"Aberdeen have said on record that they've got no money, Dundee are in the division below, which will be a hard sell for them, and I don't think Hearts have got any money.
"So, it's really down to whether Hibs want to pay money - and Darren would have to go and agree personal terms, which he's never done.
"It's been on-going for probably the last three weeks for them to agree personal terms, so I don't think Darren will be leaving this football club, to go to another team in Scotland anyway. It would be a major surprise if he goes.
"Even with everything that's been going on, I think he's played extremely well for us and nobody can ever accuse him of not trying.
"Darren's the kind of character who will do the right things.
"With certain other guys you might encourage them to go but Darren's certainly not one of them. He's proven that ever since the window has been opened and since the start of the season."
Barr was missing through suspension as Falkirk exited the Scottish Cup on Monday evening when they went down to a 1-0 defeat to Kilmarnock.
It was a blow to the Bairns, coming hot on the heels of their morale-boosting 1-1 draw with Celtic only two days previously.
However, with new signings Enoch Showunmi, Colin Healy and Jack Compton having increased May's options in his revamped squad, the Falkirk boss is convinced his players have what it takes to lift themselves away from bottom spot.
He added: "The players we've brought in are better than what we had and that has given us a boost.
"If we can get players to perform how they're capable then we've got a chance to get off the bottom of the league, which is the main priority."
Dundee Utd 11/10, Falkirk 12/5, Draw 12/5
Source: Team Talk