Birmingham have offered new contracts to four players as they continue talks with goalkeeper Darren Randolph.
Wolves' Bakary Sako has revealed he is mulling over offers from the Premier League but is yet to decide on his future.
The 20-year-old signed a pre-contract agreement with City in August and was set to arrive in the New Year after seeing out his deal with New England Revolution.
But an appeals panel rejected Stoke's application at a London hearing, and no further appeal is allowed.
Stoke chief executive Tony Scholes said: "We're bitterly disappointed that the panel rejected our appeal for a work permit for Juan.
"The criteria by which the panel should make work permit application decisions are well established and have been in place for some years and, despite recent comments to the media and discussion in the media, that criteria has not changed.
"We are therefore left amazed that our application for a work permit for Juan has been rejected when you compare his talent and ability to players who have been granted a work permit on appeal in the past.
"Unfortunately, under the rules of the appeal panel system we were given no explanation why the application was rejected and so can only speculate as to the reasons why we have been unsuccessful."