Adkins and assistant Andy Crosby were asked to take charge at St Mary's Stadium on Wednesday night, they agreed but the two clubs had yet to agree a compensation fee.
Adkins, who has been in charge at Glanford Park since November 2006, has made it clear he wants to go to the South Coast but will not leave until the Iron are happy with their financial compensation.
The Iron host Bristol City in a Championship game on Saturday, and Adkins will be in the home dugout unless an agreement is reached. Ian Baraclough, Scunthorpe's first-team coach, is expected to be made caretaker manager should Adkins and Crosby depart before the game.
Steve Wharton, the Iron chairman, criticised Southampton's pursuit of Adkins on Thursday morning, words which will not have helped his chances of getting a large fee out of his Saints counterpart Nicola Cortese.
Adkins countered Wharton's comments, praising the behaviour of both clubs.
"Southampton approached Scunthorpe, they gave me permission to speak to Southampton and I have done that," he said.
"I have done things as professionally as I can, unfortunately one or two comments got out into the media but negotiations need to be private and confidential. Everything has been done in exemplary fashion by both parties.
"If Nigel Adkins leaves Scunthorpe United it should be done in the right manner and the two clubs should be able to agree compensation. Compensation has not been agreed as it stands, negotiations were going on into the very small hours late on Wednesday night.
"One of the scenarios that could happen is that the manager has the option to resign [to force a move]. That is against my principles so that is not even being considered."
Bournemouth claimed their manager, Eddie Howe, turned down the position on Thursday - possibly in the period earlier in the day where negotiations between the Saints and Scunthorpe appeared to have broken down.