The former Middlesbrough boss was placed in interim charge back in September, after the Football Association sacked Sam Allardyce after just one game in charge.
England have won two and drawn two games since Southgate took charge and that has put them in a good position to qualify for the 2018 World Cup.
Southgate has guided England to wins over Malta and Scotland while they were also in impressive form during their 2-2 draw against Spain.
The former centre-back was interviewed last week at St George’s Park and he is thought to be the outstanding candidate with the FA willing to tie him down until Euro 2020.
Arsenal have reportedly held talks with Dennis Praet's agent and could make a £22million move for the Sampdoria midfielder this summer.