The Daily Star says Everton are willing to offer £3 million for the Northern Ireland international.
The 22-year-old recently rejected a new deal at Hillsborough and is now free to talk to the Baggies after a fee was agreed, believed to be about £3m.
Owls chief executive Kaven Walker told BBC Radio Sheffield: "We've always said that if a club matches our valuation of Chris we would not stand in his way.
"West Brom offered what we're looking for - everybody has their price."
He added: "Chris made it crystal clear to us that he does not want to be at Sheffield Wednesday.
"The manager won't get all of the money that we receive for Brunt, but he will be able to spend some of it and we're currently in talks with a couple of midfielders."