The position is vacant following Southgate's appointment as manager of the senior team on a four-year deal.
U20s coach Aidy Boothroyd had replaced Southgate on a temporary basis.
On the job advert the FA say its goal is to "win the World Cup in 2022 and sustain success thereafter".
Then-FA chairman Greg Dyke targeted winning the 2022 World Cup in Qatar when he took over in 2013, but that was then softened by Dyke's replacement Greg Clarke.
Ex-Coventry and Watford boss Boothroyd was temporarily in charge after Southgate was made interim England manager following Sam Allardyce's exit.
The team qualified for next summer's European Championships in Poland with a game to spare.
The Football Association has advertised for a new under-21s head coach to succeed Gareth Southgate - with the goal of helping the country win the 2022 World Cup.