Southgate spent eight years in charge of England and is the only manager bar 1966 World Cup winner Sir Alf Ramsey to lead the England men’s team into a major tournament final, doing so at Euro 2020 and Euro 2024.
He managed England at four major tournaments, also reaching the World Cup semi-finals in 2018 and quarter-finals in 2022.
Carsley has overseen three victories and one defeat in the Nations League since stepping up to cover the role.
Like Southgate, Carsley was previously England Under-21s boss and guided the team to the European U21 Championship in 2023.
After winning his first two games as interim England manager, Carsley tasted defeat against Greece at Wembley on Thursday before Finland were beaten 3-1 in Helsinki on Sunday.
Following the loss to Greece, Carsley said he would “hopefully be going back to the under-21s”.
After the win over Finland, the 50-year-old insisted it was “definitely” wrong to say he had ruled himself out of the running for the permanent job, but said England deserve a “world-class coach” and conceded he “is still on the path to that”.
Among the names previously linked with the permanent vacancy are Newcastle manager Eddie Howe and former Brighton and Chelsea boss Graham Potter.