The duo, who together brought to the stage John Ajvide Lindqvist's Let the Right One In, which is currently running at the Apollo Theatre, are at an early stage of negotiations, according to Baz Bamigboye.
According to the Mail, several theatre owners are also looking at securing the show for their venue, with Andrew Lloyd Webber's Really Useful Group making an early bid for it.
The play, which will focus on the boy wizard's pre-Hogwarts years, is co-produced by Rowling along with Sonia Friedman and Colin Callender.
John Tiffany's other directing credits include Once, Black Watch and he is currently Tony-nominated for his production of The Glass Menagerie on Broadway. Jack Thorne's other plays include Bunny, When You Cure Me and an adaptation of Friedrich Duerrenmatt's The Physicists.
The Harry Potter series, comprising of seven books, has sold over 450 million copies since the first one - Harry Potter and the Philosopher's Stone - was released in 1997.