The Saints have agreed personal terms with Adkins and Crosby, but a move seemed to break down on Thursday morning.
Steve Wharton, the Iron chairman, had suggested a move had broken down over the compensation the South Coast club will pay the Iron. That now appears not to be the case, with a move to St Mary's still possible.
Wharton had claimed Adkins was given an ultimatum to resign from his position as Iron manager by Wednesday night, something the South Coast club denied.
"If Nigel Adkins leaves Scunthorpe United it should be done in the right manner and the two clubs should be able to agree compensation," said Adkins.
"As compensation has not been agreed as it stands, negotiations were going on into the very small hours late last night and one of the scenarios that could happen is that the manager has n option to resign. That is against my principles so that is not being even considered."
Adkins took first-team training as scheduled on Thursday and will remain in charge at Glanford Park unless an agreement is reached between the two clubs.
"We've made sure we've trained at Scunthorpe United as professionally as we can, and i'm sure dean wilkins has prepared his team for sat as well. the speculation has been unsettling," he said.
Adkins did make it clear he would like to move to Southampton, and his move to the South Coast now seems certain.
"What a fantastic club, steeped in tradition. They've got a fantastic stadium and support base," he added.
"They've got an excellent squad of players, in the wrong half of League One. They want to get promoted straight away, they want to get promoted to the Championship, it's a fantastic opportunity for whoever goes and takes it.
"I'm very excited to have been given the opportunity to sit down and speak with the Southampton chairman."
Southampton appear to have believed they were not going to secure the services of Adkins, as they approached the Bournemouth manager Eddie Howe on Saturday.
"Southampton approached me and asked if they could speak to Eddie, he went for an interview and this morning was offered the job which he declined," the Cherries chairman Eddie Mitchell told Sky Sports News.