Spurs chairman Daniel Levy has come under fire from some supporters after Andre Villas-Boas became the seventh manager to be sacked under his stewardship.
The club's technical director Franco Baldini is understood to be keen to bring Frank De Boer to White Hart Lane, although the Ajax head coach appears keen to remain in Amsterdam at least until the end of the current season.
Former Chelsea caretaker-boss Guus Hiddink has also distanced himself from the role, while a third Dutchman, Feyenoord manager Ronald Koeman, has emerged as a possible contender.
But Redknapp, now boss of Sky Bet Championship leaders QPR after he was sacked by Spurs in June 2012, believes Sherwood should be given the opportunity to lead the club long-term.
"Tottenham do change managers an awful lot but I am pleased to see Tim Sherwood having a crack at it," he said.
"They have appointed a lot of managers in the past few years but I think the only success they had was with an Englishman.
"So let's hope Tim can get the job, he has great knowledge of the game.
"I hope they don't suddenly think, 'we must get someone because he's foreign.' They have got a boy on their books who knows the game inside out.
"He has been instrumental in helping the likes of (Steven) Caulker and (Andros) Townsend coming through. I would love to see him get the job."
The Tottenham Hotspur Supporters' Trust has demanded answers from Levy about his future plans in the wake of Villas-Boas' departure.
"Following the departure of Andre Villas-Boas on Monday morning and defeat in the Capital One Cup on Wednesday evening, the board of Tottenham Hotspur Supporters' Trust believe it is paramount that the chairman of Tottenham Hotspur Football Club, Daniel Levy, issues a statement addressing the supporters to explain the rationale behind the manager's dismissal and clearly stating his plans moving forward," read a statement released by the trust.
"We believe that now, more than at any time during ENIC's ownership of THFC, the level of frustration and anger being shown by fans renders this request fully warranted.
"As ever, it is made in the best interests of our club."
Meanwhile, Southampton's Mauricio Pochettino refused to be drawn on whether he might become Tottenham's next manager.
The Argentinian has impressed since taking over the St Mary's hotseat from Nigel Adkins but insisted he is focusing his attentions on Sunday's Premier League clash with Spurs.
"I am always very respectful with decisions people make at other clubs. It is never good when a colleague is sacked and I want to wish Andre Villas-Boas all the best," he said.
"I never talk about assumptions or rumours. We are fully focused on the game against Tottenham."
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