Southgate will reportedly finalise a deal with the FA on Wednesday with the FA, but it is thought that the news won't be made public immediately.
Following the former Middlesbrough and current England U-21 manager's interim spell, he was interviewed for the role, although he is understood to have been the only candidate.
Southgate oversaw four games as manager of England, winning two and drawing two, following the controversial sacking of Sam Allardyce.
England U-20 manager Aidy Boothroyd is likely to take over the U-21 job, should Southgate and the FA finalise a deal.