Dean Hoyle, the club's chairman-elect, will lead the search for Ternent's permanent successor - and announced his ideal candidate as a young and ambitious manager.
Bookmakers slashed their odds on Adkins being that man on Thursday afternoon, with Darlington's Dave Penney - who has said there has been no contact from the Terriers - and the former Watford boss Aidy Boothroyd also names included in the reckoning.
"We will attract the right manager, because Huddersfield Town is not hard to sell," Hoyle told Huddersfield's official website.
"There are good managers in work and good managers out of work - we'll have to see what happens. We need time to reflect, so we'll let the dust settle and see what happens.
"I think we could go for quite a while to be honest. The appointment may be made this week or in six weeks. The important thing is that we get the right man to take us forward."