The Football Association (FA) is reportedly keen to retain Southgate for the 2022 World Cup after the 48-year-old guided England to the semi-finals in Russian Federation.
Even during their morale-boosting run in Russian Federation, manager Southgate warned there was still work to do and the performance at Wembley underlined that message.
Saul (13′) quickly cancelled out Marcus Rashford's 11th-minute opener for the hosts, before Rodrigo completed the comeback win 19 minutes later.
But if the grand plan at the FA is the next World Cup in Qatar, more so than a European Championship that will conclude at Wembley in two years, it can not be said with any great certainty that Southgate will be the man entrusted with the task of taking England there.
However, the former Southampton defender remains a doubt for England's friendly against Switzerland on Tuesday.
Rashford, it turned out, was one of the players with a slightly modified role on the basis Southgate had started to suspect that, in his previous England appearances, the Manchester United striker was spending too much time too far away from the opposition goal.
"They are a top, top side and I think for spells kept the ball incredibly well, pressed us and we didn't keep the ball in the early part of the game well enough".
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"We have got 19 matches, it's not very long, but, in my opinion, we have got the best group of players in the country here".
"I'm a fighter, I'll be back soon", he wrote on social networks after the game in an attempt to reassure fans.
Southgate plans to rotate his side at the King Power Stadium in Leicester and intends to bring in reinforcements from the Under-21s squad.
England have now lost three matches in a row - against Croatia, Belgium and now Spain - proof they are still a work in progress.
"It is not a concern", he said. "But I don't think we were under any illusions, as a coaching team". We can either go back to our old style and we'll never be a top team, or we can keep trying to do what we do and suffer at times.
"He knows we have real belief in him. These are the teams we want to play against".