Southgate: Harry in contention
But former England defender Southgate maintains more work needs to be done to develop home-grown elite coaches.
Redknapp has overseen Spurs' rise up the Premier League, with Wednesday night's 2-0 home win over Everton moving the north-London club level with Manchester United and just three points behind leaders Manchester City, whom they take on at the Etihad Stadium on January 22.
Current manager Fabio Capello is set to stand down after the European Championships this summer, and Football Association chairman David Bernstein has confirmed there is a "desire" for the Italian's successor being English, although there have been no cast-iron guarantees on the issue.
Southgate last year took up a part-time role with the FA as head of elite development to work alongside Sir Trevor Brooking, the director of technical development.
And the 41-year-old former Middlesbrough manager believes an improved network of top English coaches, not just the likes of Redknapp, has to be a prerequisite for the future.
"Long-term, clearly for everybody it is preferable to have an English manager in charge of England, to support all of the coaching structure and give people a pathway through," Southgate said at Thursday morning's launch of the Carlsberg Pub Cup in London.
"It would be ridiculous to say Harry must not be a strong contender with what Tottenham are doing, but there has genuinely been no conversations about it, and the [FA] chairman [David Bernstein] has said he will make a decision after the tournament.
"What we have to do at the FA is help play our part with the coaching badges to get as many English coaches and managers qualified as high as we can.
"If you turn around and say the next England manager and then the one after that has to be English, well there is only Harry, Alan Pardew [at Newcastle] and [West Brom manager] Roy Hodgson who are working in the Premier League.
"All three of those, in different ways, have excellent credentials for what they have done.
"But we need a bigger pool of people in those positions of power."
Southgate - who won 57 international caps - believes the long-overdue National Football Centre at St George's Park, Burton-on-Trent, which is set to finally open in the summer will help become a "hub" for the next generation of managers.
"As well as being a home for the England teams, it will be coaching education and development, which is an important role because we have never had a home for coaching," said Southgate, who had a spell in charge of Boro from 2006 to 2009, but was eventually sacked following relegation from the Premier League.
"As much as anything, the interaction between people on courses is as important as what is being taught.
"If you have got courses where you might have junior international teams there, or the League Managers' Association based there, you are going to get senior managers passing through as well.
"It will be a real hub of learning and a great environment for people to go to."
Southgate also accepts the dynamics of grass-roots football is "changing", with a rapid increase in the number of people playing five-a-side at a nationwide network of purpose built complexes rather than the traditional 11-a-side format on a Sunday morning.
As to his own future, the former Crystal Palace and Aston Villa defender remains very much open to ideas.
Southgate added: "This role runs for about another eight months or so, with the FA moving to St George's Park, there will be lots of things which need help.
"I haven't ruled out at some point going back to management.
"I needed to go away and learn more, but also had to do other things after being in a football environment non-stop for 20 years - I have been able to go skiing, run a marathon and spend time with the kids.
"I quite like that diversity, but I am very open-minded about the future.
"You have to do the job your are doing as well as you can, and then whatever will be, will be."
:: Gareth Southgate was speaking at the launch of the Carlsberg Pub Cup, a nationwide five aside competition for players of all standards. To register your pub visit www.carlsberg.co.uk/pubcup Carlsberg is the official beer of the England football team and sponsor of UEFA EURO 2012.