Woodward, who was a colleague of Redknapp when the pair were at Southampton in 2005, believes that the Tottenham boss would be an ideal candidate for the post.
"He's an Englishman who's proved himself and from what I saw of him, he's ideally suited. I learned so much. The players respected him hugely," Woodward told The Independent.
"He has this incredible talent for spotting and grooming talent. I hope England do well in Euro 2012 with Fabio Capello but after that I'd like to see Harry in charge because he is a world-class coach."
Woodward is currently the British Olympic Association's Director of Elite performance.
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