Southgate, 48, guided the Three Lions to their first World Cup semi-final in 28 years and their first major tournament semi-final since Euro '96, finishing fourth at the 2018 event in Russia after being knocked out by eventual losing finalists Croatia.
The former Middlesbrough coach has since received plenty of acclaim and rumours have emerged that his name has been drawn up on a shortlist of candidates to take over at Old Trafford permanently ahead of next season, when interim boss Ole Gunnar Solskjaer is due to return to Norwegian club Molde.
However, Southgate asserted his intention to remain the England head coach ahead of the inaugural Nations League semi-finals this summer and Euro 2020.
"I'm the England manager and it's a privileged position to be in," Southgate told Sky Sports News.
"We've had the best year for 28 years and we've got a [Nations League] semi-final this summer and a European Championship [in 2020] that's predominantly at Wembley, providing we take care of business to qualify.
"So, it's a hugely exciting time with the team. I don't think we've progressed as far as we can yet. I think there's room for us to grow.
"I think there is a lot of importance in life about enjoying what you are doing.
"I am still a young coach, I've managed less than 200 matches, so, for me, it's an honour to be in the role I'm in and I thoroughly enjoy it."
Southgate also insisted that he did not want to rest on his laurels after the many tributes paid to himself and the England squad following their World Cup performances in Russia.
"When you're having tributes, you're thinking, 'Well, the job is not done for us yet, there's a lot for us to go for and a lot of improvement for us to make'."
England will take on the Netherlands in their Nations League semi-final in June after impressively topping a difficult group containing Spain and Croatia. Portugal and Switzerland will contest the other semi-final.