Chief executive Tony Scholes and manager Mark Hughes presented the club's case to an appeals panel in London on Wednesday, but were unsuccessful.
The 20-year-old's signing from New England Revolution was announced in the close-season and was set to go through in January, but the player will now be left considering his options.
Stoke currently have three Americans on their books, including Brek Shea, who successfully gained a permit to join for the 2013-14 season.
"We’re bitterly disappointed that the panel rejected our appeal for a work permit for Juan," said Scholes.
"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."
Colombian-born Agudelo has 17 caps for the US national side.
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