England are at the bottom of League A Group 3 in the Nations League with one point after losing 1-0 to Hungary and drawing 1-1 with Germany. They face Italy on Friday, who are top with four points.
Southgate, who led England to the 2018 World Cup semi-finals and the Euro 2020 final in 2021, signed a new contract through to the end of 2024 in November last year.
"I won't outstay my welcome but I think I can do a good job for the team, and I think we have done a good job for the team," Southgate told reporters.
"If you look back I have always said it. I am not going to be here forever, am I? There will be another England manager.
"The major part of that is how the players are. Do I still feel the players give everything and respond to what we do? Yes, I do."
Southgate has often been criticized for being too cautious, with pundits and fans urging him to play Liverpool's Trent Alexander-Arnold and Manchester City's Jack Grealish more often.
"... I hear about conservatism, but we were the highest scorers in Europe last year," said Southgate, referring to England's 39-goal haul in the World Cup qualifiers, their highest tally in a qualifying campaign.
"I am always going to have those situations where people disagree with the selection. That noise over the last two or three years does seem extremely loud and I've had to ride that and get on with it through the Euros and the whole of last summer."
Following their game against Italy later on Friday, England will host Hungary on June 15.
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