England were outclassed by Spain as La Roja won 2-1 on the night.
Southgate is coming off contract and he said: "I totally understand the question and understand you need to ask it, but I need to have those conversations with important people behind the scenes and I'm obviously not going to discuss that publicly first.
"Without a doubt, England have got some fabulous young players. We have now been consistently back in the matches that matter. It's the last step that we haven't been able to do.
"It's hard to reflect so soon after a defeat like this.
"Of course, to take England to two finals has never been done but we came here to win and we haven't been able to do that."
Southgate blamed fatigue on their flat final performance.
"I think in the main, at the end of the game, the physical issues that we had probably took their toll," Southgate said.
"But at the moment, I have to say, in my head none of that matters because we had an opportunity to win and we haven't been able to take it."