The young footballer, who spent the last season on loan at Leeds United, is now returning to the Championship.
This time he will be playing for Clarets, who were relegated from the Premier League last term.
Anthony had hoped that he could come back to the Cherries and impress this summer.
However, the 24-year-old has not been used regularly by manager Andoni Iraola.
He will now hope to resurrect his career away from the club and then earn a permanent exit.