Nigel Adkins, the Scunthorpe boss, was offered the job on Wednesday night but the South Coast club and the Iron have yet to agree a compensation deal.
Steve Wharton, the chairman at Glanford Park, criticised Southampton after they urged him to resign from his position to strengthen their hand.
He refused to walk out on a club he has been employed by for 14 years, prompting them to offer the job to Bournemouth's Eddie Howe - who turned it down.
It is unclear where Adkins will be on Saturday, but the fact he took training on Friday morning suggests he will oversee Scunthorpe's Championship game against Bristol City in what will likely be his final Iron game.
That would mean Dean Wilkins continuing as caretaker Saints manager against Wilson's Swindon side at the County Ground.
"There's no room for mistakes. They have to go in there straight away and get results, that is the pressure that comes with the size of that job," Wilson told the Swindon Advertiser.
"It is a fantastic football club, a big football club, and whoever goes in there will not only be in charge of an ambitious club but the expectation level is sky-high."
Southampton raised eyebrows when they sacked Alan Pardew so soon into the season, and their pursuit of Adkins drew stinging criticism from Wharton on Thursday.
The Saints deny making any ultimatum or withdrawing their offer to Adkins, but it is hard to see why they offered the position to Howe if they had not turned their attention elsewhere.
"It's crazy - absolutely crazy. If you're going to make a decision then make it in the summertime," added Wilson.
"Why wait until this time if you're not going to get on with your manager or whatever reason it was? What is the point in waiting around, because this is what happens - it unsettles them.
"I think there is a better and more sensible way of doing things, but that's the way some people work and it's not my place to tell them how to run their business and their clubs."