The Daily Mail says Saints chairman Nicola Cortese hopes to reveal Alan Pardew's successor later this week in the wake of Saturday's home defeat by Rochdale.
The next manager will be Southampton's 13th of the millennium. Cortese has been impressed by the way Adkins led Scunthorpe into the Championship and kept them in the division last season despite having the division's smallest gates.
Adkins was appointed caretaker manager in November 2006, following the departure of Brian Laws. Adkins had previously held the role of physio at the club and was appointed permanent manager the following month.