Stoke originally signed Agudelo last summer on a pre-contract from MLS side New England Revolution ahead of a move in January - but in November he was denied a work permit.
Stoke have signed Agudelo anyway, though, confident they can win the 21-year-old a permit in March when they can reapply. They will now loan him out to a club in Europe where there are less restrictions.
The player Tweeted on Tuesday evening: "There's no other club that has supported and believed in me like @stokecity FC has. #SignedToday - 6 month loan imminent."
Agudelo's announcement came just hours after Stoke finalised the permanent capture of Stephen Ireland from Aston Villa and a loan move for Manchester City forward John Guidetti.