Miguel Herrera has been fired as Mexico coach following allegations that he punched a television reporter.
Former Arsenal midfielder Abou Diaby has rejected an offer to join West Brom in favour of finding a new club in France.
Edu has signed a deal after making only one first-team appearance for Stoke - as an 80th-minute substitute - since joining them from Rangers in August.
Potters boss Pulis said: "He has come in and not got any games.
"That has been disappointing for Maurice and he needs to play games, to maintain his opportunity of playing for his country.
"So the club (Bursaspor) has come in and it is a decent club. We have had offers in England as well for Maurice to go and play, so he has got one or two options, which is good.
"Having bought him, and then for him to have then not gotten into the team, we can actually monitor him now for the next three or four months."