The former Sheffield United coach, who was sacked for Chris Wilder earlier this term, wants to get back into management.
However, The Sun states that Heckingbottom may be told he can only bring in one backroom staff member if he joins the Black Cats.
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He would have to pick between out-of-work Stuart McCall or Sheffield United coach Jack Lester.
Sunderland want to hold onto most of their existing coaching staff for the coming season.
Besides Heckingbottom, they are also assessing Sheffield Wednesday boss Danny Rohl.