Adkins has clearly interested the League One Saints as they look to replace Alan Pardew, with Sky Sports News going so far as to say he has been given permission by the Iron to speak to the South Coast club on Tuesday night.
That was followed by the Southern Daily Echo reporting on Wednesday afternoon that he was expected at the club for talks, who were negotiating compensation terms with the Iron.
Adkinswas quickly made one of the bookmakers favourites, with the former Hull City boss Phil Brown also believed to be a strong contender after revealing on Wednesday he has held talks with the club.
Victor Chandler stopped taking bets on Adkins on Wednesday morning after taking odds-on bets as high as £5,000 on him being installed at the St Mary's Stadium.
On Tuesday the highest bets they had taken was £500, prompting them to cut his odds drastically.
Adkins and the Iron chairman Steve Wharton have refused to comment on the suggestions, perhaps significantly not taking the opportunity to rule him out of the running.
Adkins said on Tuesday that his only interest was in preparing the Iron's first-team to face Bristol City on Saturday.
"I am totally focused on Scunthorpe United - the wheels are already in motion preparing for our home game with Bristol City on Saturday - the first of seven we have before the next international break," Adkins told the Scunthorpe Telegraph, comments printed in their Wednesday edition.
"I have been here at Scunthorpe now for 14 years. I think people can see how hard I work and I would like to think that rubs off on everybody else here.
"I am ambitious and there will come a day when I walk out of here, hopefully because of what we have achieved here - but who knows when that will be?"