England is winless in five matches – its worst such run since June 2014 – ahead of taking on historic rival Germany at a sold-out Wembley in its final UEFA Nations League tie.
The Three Lions were relegated from Group A3 after Friday's 1-0 loss to Italy, which followed a 4-0 hammering at the hands of Hungary in their most recent home game.
Southgate was booed after both of those defeats, with fans also heard chanting "You don't know what you're doing" during the contest against Italy at San Siro.
But ahead of England's first game at Wembley since March, in what is their last outing before the World Cup, Southgate hopes his players are backed by those inside the ground.
"We've got 90,000 people – the stadium is sold out – so people want to come and see this team play," he said at Sunday's pre-match press conference.
"That's because the players have done an unbelievable job for six years.
"We were on the back of a difficult time in terms of the relationship with the fans at the start of that journey and slowly we've built with the finishes that we've already discussed.
"It is not healthy for the team to be having this noise around them. I fully understand that. But it is for me to take responsibility, it is for me to allow them to go and play.
"I want them to feel freedom. They know we always talk about that around the training ground, on the training pitch, and I would urge the supporters to get behind the team.
"How they deal with me at the end or whenever, on the phone-ins or wherever else is completely different.
"But this is their last chance to see the boys before they go to a World Cup and we are all in it together. We can only succeed if we're all pushing in the same direction.
"What happens to me is irrelevant, frankly. It is about the team. The most important thing is the team and the success of the team."
Across his six years in charge, Southgate has guided England to the 2018 World Cup semi-finals and Euro 2020 final, as well as reaching the Nations League Finals in 2019.
England has won just two of its seven matches this year, however, with both of those coming in friendlies, and have failed to score from open play in more than eight hours.
Indeed, the Three Lions are one of only two nations yet to score a single non-penalty goal in the 2022-23 Nations League alongside minnows San Marino.
Asked how he is coping with the recent criticism, Southgate said: "You're not going to have six years as we've had without a spell where you are going to have some tough results.
"You've got to show resilience to come through those moments. I'm not the first coach to go through a difficult time in terms of results and criticism.
"This is part of the territory – for me, it's a great challenge to lead the team through a moment like this.
"The results haven't been at the level we want or require. So, no matter what job you have in football, that would be the case.
"Of course, with the national team, that noise is going to be louder and more widespread, I understand that."