The Black Cats had a bid rejected that would have seen Gradel involved in a similar deal to one proposed by Toulouse.
There was also interest from ex-club Leeds, where Gradel worked with Simon Grayson before leaving for St Etienne.
Hull also registered interest in the last transfer window.
But Gradel has looked at options abroad as thrashing out a deal to stay in England has proved difficult.
He only signed a new four-year deal a year ago at Dean Court after returning from a knee injury and impressing with his effervescent performances.
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Gradel was supposed to be in Eddie Howe’s plans last season but all of his Premier League appearances came off the bench.
But Howe has been impressed with him in pre-season and will now consider whether he wants to let him leave.
Bournemouth player Max Gradel discusses his relationship with football
Gradel penned a four-year-deal after recovering from injuryGetty ImagesHe impressed boss Eddie Howe... but not enough to start Premier League gamesNews Group Newspapers Ltd
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