The English pair come head to head when Tottenham Hotspur host Sunderland at White Hart Lane on Saturday.
Redknapp has been mentioned as a contender for the England job himself and he feels that it is time for an Englishman such as Bruce to be considered more seriously for the role.
Pride of England: Harry Redknapp (left) and Steve Bruce
Bruce has been earning plaudits for guiding Sunderland up to eighth place - just two points behind Tottenham, who are fifth.
Former Manchester United centre back Bruce took over at the Stadium of Light this summer after leaving Wigan Athletic and he has previously had spells in charge of Sheffield United, Huddersfield, Crystal Palace and Birmingham City during a decade in management.
Redknapp said: 'He's a great guy and a good manager. When the job comes around again, he's the type of lad who should be right up there, pushed forward for it.
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'There's no reason he can't go on to be England manager, for sure. His teams move the ball, pass the ball and he has always got the best out of the teams that he's had.'
Redknapp knows that Bruce's Sunderland side will be highly motivated because of the strong ex-Tottenham contingent that they have there.
'This year, I think they'll finish in the top half,' said the former West Ham boss, dismissing suggestions that the Weirsiders could be sucked into another relegation battle later on in this campaign.
'They'll have three or four ex-Tottenham players playing against us so they'll look forward to the game but I believe that we can beat anybody at home.'
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