Southgate's future as Three Lions manager remains up in the air after England's 2-1 defeat to France in the World Cup quarter-final.
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The 52-year-old confirmed there will be talks with the FA and a decision will be made in due course.
Southgate has a contract until 2024, but his future appears to be up for debate and it has divided opinion on whether he should stay or go.
Former Tottenham boss Redknapp, who was linked with the England job back in 2012, believes Southgate should remain as Three Lions manager given he boasts a young squad.
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England have Jude Bellingham (19), Bukayo Saka (21), Phil Foden (22), Mason Mount (23), Declan Rice (23), and Marcus Rashford (25) in their ranks, so the future looks very exciting.
"He [Gareth Southgate] must look and think 'it must be crazy to walk away with so much talent at my disposal'," Redknapp told talkSPORT.
"I am never going to get this opportunity again, and must win something before I pack this job in."
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Ex-Tottenham and Paris Saint-Germain boss Mauricio Pochettino, and former Chelsea manager Thomas Tuchel have recently been linked with the job.
Redknapp, though, insists it should be an Englishman to take the group of players forward.
"I would like to see an Englishman manage England for sure," he added.
"Henry Winter just now, was saying all the lads do their coaching badges and go to St George's Park, we have lots of managers who are coming through the system.
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"I think with this group of players, I really do believe there is a tournament to be won."