Southgate was interviewed by a five-man panel at the St George's Park national football centre last Monday.
Southgate, who recently completed a four-game stint as interim manager, was interviewed by a five-man panel at the St George's Park national football centre in Burton-on-Trent, central England, last Monday.
Southgate is the only candidate to have been interviewed for the post and newspapers including The Daily Telegraph, The Guardian and The Sun said he had been offered the job.
England's Football Association has not commented on the process.
Southgate, 46, is thought to have been offered a four-year contract, which would take him through to the 2020 European Championship.
The former Middlesbrough manager stepped up from his role as England Under-21s coach after Sam Allardyce left his post in the wake of a newspaper sting.
Southgate oversaw two wins and two draws in his four games at the helm, culminating in a 2-2 friendly draw against Spain at Wembley last week.