The 46-year-old took over the role on a temporary basis when Allardyce was sacked, and after overseeing two wins and two draws, he became the leading candidate to seal the job.
"The biggest challenge is to be successful at a tournament," Mills told BBC Radio 5 live. "He has the opportunity now with younger players that he understands and he has worked with before. He is more progressive than other managers. He can get to understand some individuals as people as well as players.
"We travelled a lot together so I maybe I am biased. He has direct principals on how he wants to play football. He was compromised at Middlesbrough because he had to sell players.
"There will be a mix of good football and professionalism. Ultimately he will want to win. He is one of the nicest guys in football but he has an edge. You don't win 50-odd caps and captain your club without having something about you."
Southgate's first game as permanent manager will be an international friendly against Germany on March 22.