Former Tottenham and Paris Saint-German manager Pochettino has said he would be open to succeeding Southgate, while ex-Chelsea boss Tuchel is also rumoured to want the role.
Southgate’s future beyond the World Cup is unclear and there has been speculation this could be his final tournament in charge.
Ahead of Monday’s Group B opener against Iran, Southgate says interest from top-level club managers is a result of England “making the impossible look possible”.
“Maybe we’ve made the impossible just look possible,” said Southgate. “We’ve made it look possible [to win] and it’s exciting for other people and I can understand that.”
Southgate guided England to the semi-finals of the 2018 World Cup and the final at Euro 2020. But England go into the World Cup on a six game winless run and Southgate is under scrutiny in Qatar.
“I think in between all of the tournaments we have looked at different ways that might be the right way, might not be the right way, individuals, see how they might fit in, what they might bring to the group,” said Southgate.
“Of course, the results in the summer were the thing that created the tension around September in particular. But Germany was the game we highlighted to put as stronger a team out as possible.
“The rest, it was bizarre really. We were playing international football but managing minutes like a pre-season and trying to avoid important players getting injured.
“We had to accept the criticism that came with that and it is understandable. We don’t lose a home game 4-0 and expect to walk off without criticism of the result and the performance. That’s my responsibility, I have to do that.
“But it is also my responsibility to remind the players that actually there is a lot they’ve done well and coming into this tournament, they shouldn’t be focusing on the recent past.
“It is hard to talk about form because you are always six weeks, two months between games in international football so what actually is form?”
There is belief in the England squad that they can go all the way in Qatar but Southgate admits the team is not yet where they want to be.
“What there is, is pedigree,” he said. “We want to be a Germany, who when I was looking at their Wikipedia page: four golds, four silvers, four bronzes, European Championships three goals, three silvers, three bronzes.
“Yeah, our page didn’t quite look like that but we’d love it if it did in 40 years’ time and that should be our aim, to be consistently challenging.”
