Huw Jenkins, the Swans chairman, is leading the search for a replacement for Paulo Sousa, who left the club last week to take charge at Leicester.
"We are still working to finalise a few details, over the next few days we should be able to get our manager in," he told Sky Sports News.
"It has proved a little more difficult than we thought but that's because the people we are after are well thought of in football.
"I'd like to think we can have something by the weekend and will work night and day to make sure we can do that and get things settled down again."
The Welsh club have made three official approaches: at least two of which have been turned down. None are believed to be for Adkins.
The first was for Gary Speed, who is coaching at Sheffield United. The Blades rejected that approach.
Billy Reid, in charge of the Scottish Premier League side Hamilton Academicals, was the second but that was also turned down after Reid said he wanted to stay north of the border. The third is Exeter City's Paul Tisdale.
Adkins remains one of the favourites with the bookmakers, even though the Swans chairman Huw Jenkins has repeatedly said Adkins' loyalty to Scunthorpe makes that unlikely.
Graeme Jones, who was assistant to Robert Martinez at Swansea, has been linked with a return but appears unlikely to leave Wigan where he continues to work with Martinez. Brendan Rodgers, former Watford and Reading coach, has also emerged as a contender.