The Potters have parted ways with Alex Neil following a poor run of results in the Championship.
They are on the hunt for a new manager to come in and get more out of the players at the club after Neil was heavily backed in the summer transfer window.
Stoke are now in the process of checking out the situations of managers that they admire, as they prepare to act.
And Eustace is likely to have a strong chance of being in the mix after his good work at Birmingham.
Birmingham raised eyebrows by sacking Eustace to make way for Wayne Rooney, who has struggled badly since taking charge.
Eustace’s stock remains high and Stoke may feel he is the right man to come in, just in time for the opening of the January transfer window.
Stoke are due to pay a visit to Birmingham on Boxing Day and it remains to be seen if Eustace might be at the helm when they take on Blues.
Birmingham have slumped down to a lowly 17th with Rooney in charge.
