The 45-year-old, from Fewston near Harrogate, was installed as the bookmakers' early favourite to take the job immediately after Hodgson resigned following Monday’s humiliating Euro 2016 last-16 exit to Iceland.
Football Association chief executive Martin Glenn suggested on Tuesday it could take as long as a year to name a permanent successor, saying Under-21s manager Southgate would be a “pretty obvious” interim solution.
However, it was widely reported on Wednesday evening that Southgate has no interest in succeeding Hodgson, nor has he been contacted by the FA.
Those reports saw his odds drift, although the man that led the Under-21s to Toulon Tournament glory last month remains favourite ahead of ex-England manager Glenn Hoddle, United States boss Jurgen Klinsmann and Sunderland’s Sam Allardyce.
FA chief executive Glenn, vice-chairman David Gill and technical director Dan Ashworth are to meet later this week to discuss the role further.
Glenn said the FA would not appoint an interim manager if he felt it would jeopardise qualification for the World Cup and conceded it was “unlikely” the FA would wait for a year, but admitted it was not inconceivable.
“When we get them done, we will share them with you,” he said..
“It is not 24 hours since we got knocked out of the tournament. I don’t think it is reasonable to have every ‘i’ dotted and ‘t’ crossed on this one.
“But clearly I think you want them to be an inspirational manager who is sophisticated enough to harness all the modern techniques that you can do to build resilience in tournaments and overall success.
“I think it is somebody who has got clearly the track record but also the breadth of experience not to just stand on the sideline and holler, but to plan.
“And that is why we are looking for a manager - and a wider team - to support them.”