However, Southgate has a contract until 2020, which could be extended until beyond the 2022 World Cup, giving England stability.
"We've said now that at every age group we want to be constantly challenging for finals because if you're in those moments of tournaments, your chance of success are high", Southgate added.
Mrs May was unable to watch the game against Croatia, which saw England crash out of the competition, due to a North Atlantic Treaty Organisation summit but insisted everyone should be proud of what England achieved.
Southgate said: "We have a captain whose first thought all tournament has been how can he help the team to progress".
Wembley will host several matches - including the semi-finals and final - as Euro 2020 takes place across various European cities, and Southgate said: "We nearly have a home tournament, it's going to be brilliant".
England will return to their Repino base near Saint Petersburg ahead of tomorrow's (10pm Brunei time) third place play-off against Belgium, a repeat of their final group match.
The 2018 FIFA World Cup kicked off on June 14, with the final set to be held at Moscow's Luzhniki Stadium on Sunday. We've one or two injuries, one or two with a little bit of illness, so we just have to see.
England now holds the under-17 and under-20 World Cup crowns.
France's lost Euro 2016 final a driving force for coach Deschamps
By no means are they assured a spot, as world No. 7 France , who oddsmakers are favouring, stand in their way. We don't realise the support we're receiving like we did at the European Championships.
A gut-wrenching 2-1 loss to Croatia after extra-time denied England a shot at a second World Cup triumph, instead setting up a consolation game against familiar foes Belgium.
The two sides must pick themselves up and dust themselves down after suffering the crushing disappointment of semi-final defeats to play for the modest prize of third place in a fixture often criticised as pointless.
"It's a young team; the majority of the players will play in the next World Cup with more experience at club level", the Portuguese boss told RT. We are all friends and I want all the best for them.
The Three Lions were just 20 minutes from their first final since 1966, but went on to lose 2-1 in extra time.
"We need to regroup and recover and see the opportunity".
Regarding the expectations of the Croatians for the success of their selection at the 2018 World Cup in Russian Federation, he indicated that "I know what our people expect, how many people love Croatian football and our team".
Manchester United manager José Mourinho also praised the England team. "If we win, we can look back on it with a lot of pleasure, and that is what it is all about".