The Three Lions were beaten on penalties by Italy in the Euro 2020 final at Wembley on Sunday night.
Football Association chief executive Mark Bullingham said during the tournament that he wants to hand Southgate a new deal.
But Southgate told a press conference on Monday: “I don’t think now is an appropriate time to think about anything. We have got to qualify for Qatar but I need some time to go away and watch the game and reflect on the whole tournament, I need a rest.
“It is amazing experience but to lead your country in these tournaments takes its toll and I need a break now.”
Southgate has said he wants to lead the team to the 2022 World Cup in Qatar.
The 50-year-old continued: “I said at the time it is great to have that internal support, you greatly value that as a manager.
“I don’t want to commit to anything longer than I should and I don’t want to outstay my welcome so all of those things need consideration.
“I want to take the team to Qatar, I feel we have made progress over the four years, we have had a fourth, a third and a second-placed finish and that is as good as anyone.”
The manager said it was too early to be able to reflect on the positives of the journey to the final.
“I know in time we will have an appreciation of that and I know the messages we are getting underline that,” continued Southgate.
“But when you are in sport and you get to a final you know those opportunities are rare and to be so close and to know what that has taken and know you have to pick up and go again, that is very hard the day after. We will go again of course.”
Southgate rejected the theory that England are too nice to win major tournaments.
“I don’t see that really, we got to a final for the first time in 55 years,” he said. “I am not really sure...you’re not a nice team to get to a final. We have overcome a lot of hurdles that we have been unable to in the past.
“If we had been torn apart or been a mess then I might look at it differently. It is easy to throw theories around at the end of any tournament, but it is important to analyse correctly when you are a coach, I don’t want to go along with theories that I don’t think are correct.”