England lost in the final of the tournament for the second time in a row under Southgate after substitute Mikel Oyarzabal scored an 86th-minute goal to give Spain a 2-1 victory.
"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," Southgate told reporters.
"Without a doubt, England have got some fabulous young players," he said. "We have now been consistently back in the matches that matter. It's the last step that we haven't been able to do."
England - with only the 1966 World Cup to their name as a major trophy - have become a force in tournaments since Southgate took over the team at a low ebb in 2016.
They reached a semi-final and a quarter-final at the last two World Cups as well as the final of Euro 2020 - when they lost to Italy on penalties - and 2024.
"It's hard to reflect so soon after a defeat like this," Southgate said when asked if he would be in charge of England for the 2026 World Cup.
"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 congratulated Spain and said they were the best team of the tournament.
He pointed at fatigue and injuries as a reason England were unable to match Spain's control of much of the match or build on the momentum of Cole Palmer's 73rd-minute equaliser after Nico Williams had put Spain ahead two minutes into the second half.
"I think in the main, at the end of the game, the physical issues that we had probably took their toll," Southgate said.
He spoke of his pride in the "incredible run" of his players to reach two European Championship finals back-to-back.
"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," Southgate added.
