The Three Lions were defeated 2-1 by Spain in the Euro 2024 final in Berlin on Sunday.
Southgate, 53, has been at the helm since 2016, but his current contract is set to expire in December.
When asked about his future, Southgate told the BBC: “Now is not the time for me to speak about that. I need to talk to the right people and give myself a bit of time.
“To get to another final… it was a privilege to have the opportunity. But to come up short is hard at the moment.”
Under Southgate’s leadership, England has reached consecutive European finals, the semi-finals of the 2018 World Cup, and the quarter-finals of the 2022 World Cup.
Despite England progressing to the knockout stages as group winners, Southgate faced criticism in Germany this month. Supporters threw plastic cups at him following their 0-0 draw with Slovenia in the group stage.
“The players will take enormous credit for getting us to where we did but when you’re as close as that, you have to take your chance,” Southgate told ITV.
“They have represented the shirt with pride and haven’t been beaten until the very end. I just think Spain had more control of the game.
“England are in a really good position in terms of the experience they have. Most of this squad will be around for the next World Cup and maybe the next Euros.
“There’s a lot to look forward to but at this moment, it’s not any consolation.”
Southgate admitted it was “hard to reflect” on his eight years in charge but has no doubts England can be successful in future major tournaments.
“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,” he added.
“England have some fabulous players. We have been consistently back in the matches that matter. It’s the last step we haven’t been able to do.”
‘The FA has a big decision to make’
England captain Harry Kane expressed strong support for manager Gareth Southgate, stating that the players “love the manager” and were deeply motivated to win the final for him, especially after their defeat against Italy three years ago.
England has now become the first nation in Euros history to lose consecutive finals. Southgate is also the first manager to end up on the losing side in two finals.
“I suspect it will be Southgate’s last game,” former England striker Alan Shearer told BBC One.
“There will be great disappointment at home. They could have played more attacking football but they got to the final. But when you are there, you have got to get over the line.
“The reality is, Southgate got us to a final three years ago and got us to a final here, and hasn’t won. That will hurt him and maybe he will think it’s time for someone else.”
Ex-England striker Gary Lineker, speaking on BBC One, said he suspects Southgate “may have had enough” from a “tough, gruelling job”.
Meanwhile, former defender Rio Ferdinand says Southgate “has to take some stick” for England’s defeats in the finals.
“You have to look at what talent pool we have and I don’t feel like we are putting them in the best position possible,” added Ferdinand.
“The Football Association (FA) has got a big decision to make.”