Before Capello's exit on Wednesday, Redknapp was already favourite to take over from the Italian after Euro 2012 and now, with a major championship four months away, his name is on the lips of many pundits and fans.
"Our position has been clear - the fans want Harry to stay at Tottenham, the players and the coaches do, the board does, and we hope he does stay," Spurs director Keith Mills told BBC radio.
"If he is [approached] then Harry clearly has a big decision to make. He's managing a top-three team in the biggest league in the world, we're doing incredibly well at the moment. I know he loves it, it will be a tough decision for him."
Capello's departure in the wake of a row over the deposed England captain John Terry came on a day when Redknapp was cleared of tax evasion charges after a 13-day trial.
Having left court a free man with an unblemished record, he soon found himself in the spotlight once again as the media circus launched into full flow following the shock resignation of the England manager.
The front and back pages of British newspapers were full of speculation over what would happen next with The Sun declaring "'Arryvederci" and "We want Harry" and The Daily Telegraph stating "Capello Out. Redknapp in?"
It was uncomfortable reading for Spurs fans, who have enjoyed good times under Redknapp, who led the team to the Champions League quarter-finals last season.
This season they are challenging for the Premier League title, sitting in third place with 50 points from 24 games, five points behind second-placed Manchester United and a further two behind leaders Manchester City.
"I just think at this stage it is very premature to talk about Harry's departure from Tottenham," Mills said, pointing to Redknapp's contract which runs until 2013.
"He is an extraordinary manager, he's done great things at the club and there's been speculation to be frank for several months because we knew that Fabio would be leaving after the Euro competition anyway."
Asked what he thought of the possibility of Redknapp taking the England job on a caretaker basis while staying at Spurs until the end of the season, Mills said: "It's difficult to comment on hypothetical situations.
"Clearly that's not an ideal scenario. I think a football manager needs to focus on doing the job in hand and right now I think Harry, apart from recovering from an ordeal he and his family have been through for the last five years through the courts, he's probably focusing more on the Newcastle game on Saturday.
"I think trying to do two jobs is probably going to be a bit of a stretch for anyone."
Even British Prime Minister David Cameron weighed into the debate, saying he was sorry to see Capello go.
"England now needs a new coach and a new captain and I hope we can get on with that and make sure we make the best of the opportunities this summer," Cameron told reporters in Stockholm.
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