Pochettino, who left Chelsea by mutual consent on Tuesday, has a long-standing interest in the England job.
If Southgate’s team does not win the Euros in Germany this summer, the job could become available.
Pochettino, currently unemployed, is also a potential candidate for Manchester United if they decide to part ways with Erik ten Hag after the FA Cup final.
Additionally, Saudi Arabian club Al-Ittihad might show interest in him through their sporting director, Ramon Planes, who worked with Pochettino at Espanyol.
Pochettino received a pay-off from Chelsea, but it is unclear if he can immediately take on a new role.
He has previously expressed his desire to manage England, emphasizing his strong ties with English football and his commitment to developing young talent.
Football Association technical director John McDermott, who has worked closely with Pochettino at Tottenham Hotspur, would play a key role in selecting Southgate’s successor. Pochettino is known for developing English players, including Chelsea’s Cole Palmer.
England are one of the favourites to win Euro 2024, alongside Germany and France.
However, they have not won a major trophy since the 1966 World Cup. Another disappointment could lead to Southgate stepping down.
Southgate, who has managed England since 2016, has achieved significant milestones, including reaching the World Cup semi-final in 2018, the Euro 2020 final, and the World Cup quarter-finals in 2022.