Martin is expected to depart Swansea to take up the post at relegated Southampton, with Saints keen to return to the top flight at the first time of asking.
The Welsh giants have been drawn to Eustace as an option after being impressed with his work at Birmingham.
It is unclear though whether Eustace would be open to leaving Birmingham to take charge at Swansea.
Birmingham finished 17th in the Championship this season, keeping well clear of the drop zone and securing their status for next season.
Swansea meanwhile came tenth under Martin, six points off the final playoff spot.
Eustace, 43, took over at Birmingham last year in July and put pen to paper to a three-year deal at St Andrew’s.
All eyes will be on whether Swansea now firm up their interest in Eustace and make an approach to Birmingham to hold talks with him about a move to Wales.
