The 46-year-old has steadied the ship since being parachuted in following Sam Allardyce's abrupt exit, ensuring the Three Lions ended a turbulent year top of their World Cup qualifying group.
While England were caught cold by a late Spain comeback as his four-match temporary stint came to an end, the promising display in Tuesday's 2-2 draw will only have strengthened Southgate's grip on the job.
Southgate believes such conversations are best had in private as the FA's formal process into selecting the next permanent England manager gets under way.
"Not a great deal," the England Under-21s manager said when asked what he can say about his future.
"I have a couple of days off now to go and reflect and review the games and report back on everything I've seen and been involved in.
"I will reflect on the immense pride of leading the team and leading the country at a time where there was quite a bit of instability.
"I feel we've managed to more than stabilise the group. I think they are on a good platform now to push forward.
"I've been proud to lead and fulfil the remit I was given for these four matches.
"The rest we will just have to see what emerges over the next few weeks."
FA technical director Dan Ashworth, chairman Greg Clarke and chief executive Martin Glenn will be part of the panel that selects the next manager.
League Managers Association chairman Howard Wilkinson will form part of that decision-making process and it is understood so will Graeme Le Saux.
A former international team-mate of Southgate, the 48-year-old has long been connected to the FA in a variety of roles after a career that saw him win 36 caps for his country.
One bookmaker's odds of 1-50 highlight just how much of a formality Southgate's appointment appears to be, having impressed the FA in the way he has handled a job he did not feel ready for when Roy Hodgson left in the summer.
Friday's 3-0 defeat of oldest foes Scotland was the most memorable moment of his unbeaten four-match stint, although there were arguably more promising signs against Spain.
After the match, Southgate refused to say he wanted the England job on a permanent basis despite insisting it proved he could "manage big matches".
The 46-year-old said: "I've learned a lot about the role. I've proved to myself I can manage big matches, that I can tactically - with the help of my staff - prepare the team to play a top side and give them a really tough test.
"I feel every situation we've been through - many of which have been complex, and potential powderkegs - we've coped with very well.
"This has been a brilliant experience, I've thoroughly enjoyed working with the players; I can see the potential of the group.
"When I took over it was a moment of instability for everybody. We've brought stability back and built a platform for the team to push forward. In terms of the remit we were asked to do, we've fulfilled that.
"I'm proud to have led the country for a couple of really important games and experienced what that feels like."