The forward was linked with a move to Celtic Park in January, but signed a new contract at Villa before being loaned to Hull City.
However, Villa are now reportedly prepared to sell Barry before Monday to avoid breaching the Premier League's PSR rules, according to HullLive.
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News of his potential exit has reportedly caught the attention of several clubs in the English Championship.
Celtic were reportedly ready to bid a seven-figure sum for Barry in January.
It's not yet clear if they'll make a move for the 22-year-old this summer.
Meanwhile, Shaun Maloney has taken on the role of Celtic's new Professional Player Pathway Manager, succeeding Darren O'Dea.
Maloney said: "It is a huge honour to be given this opportunity to return to Celtic.
"My aim will be to work really closely between our Academy and the First Team and create the very best opportunities for success.
"This will ultimately be represented by more and more players reaching our first team environment.
"I have been fortunate enough to follow this same pathway, and I am now excited about ensuring many more young players can take this same journey."