And Coleman is interested in taking the job with the Championship’s bottom club, according to the Daily Mail.
The 47-year-old’s future as Wales manager is uncertain following their failure to reach next summer’s World Cup.
Coleman is out of contract but FAW chiefs have attempted to extend his contract until 2020.
Sunderland have been on the hunt for a new boss since axing Grayson on October 31.
Bain hopes to appoint a new manager before Saturday’s home clash with Millwall but is not prepared to rush such a crucial decision.
Caretaker manager Robbie Stockdale will continue against Millwall if an appointment is not made by the time the Lions visit the Stadium of Light.
Sunderland will set a new English record of 20 home games without a win if they fail to beat Neil Harris’ side.