Southgate is in talks with the FA about extending his contract, which expires after the Qatar World Cup, and Aston Villa defender Mings hopes the England boss agrees a deal.
He told BBC Radio 5 Live: "I would welcome that every day of the week.
"The opportunities he gives you and the belief he gives you is very special."
England were unbeaten in qualifying, finishing six points ahead of play-off bound Poland, with eight wins from 10 matches.
Southgate has previously led the side to a World Cup semi-final and a European Championship final, after succeeding Sam Allardyce in 2016.
"What he has created in this England set up is very special", added Mings. "He has been a fantastic manager for me and a fantastic manager for England.
"I don't need to speak too much about him because his record speaks for itself at tournaments. I would love to see him carry on as manager."
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