"I have thoroughly enjoyed my time at Spurs and am proud of my achievements," Redknapp said on the
"I have had a fantastic four years with the club, at times the football has been breathtaking. I am sad to be leaving but wish to thank the players, staff and fans for their terrific support during my time there."
Everton manager David Moyes has been installed as one of the early favourites, along with Rafael Benitez, Roberto Martinez, Andre Villas-Boas and ex-Spurs striker Jurgen Klinsmann, currently coach of the United States.
Levy thanked Redknapp in a statement on the club's official website, which read: "This is not a decision the board and I have taken lightly. Harry arrived at the club at a time when his experience and approach was exactly what was needed.
"This decision in no way detracts from the excellent work Harry has done during his time with the Club and I should like to thank him for his achievements and contribution. Harry will always be welcome at the Lane."
When Capello left his post as England boss on 8 February, Spurs were third in the league, 10 points clear of a faltering Arsenal but they eventually finished a point behind the Gunners.
Levy is understood to have been left disappointed by Spurs' end-of-season slump, and is said to be in no rush to replace Redknapp with the club privately expressing a determination to conduct a "properly run process" to find their next manger.
It is understood no contact has been made with either with Everton or Moyes, who has been on holiday in the US with his family and is said to have been unaware of the overnight speculation linking him with the Tottenham job as he arrived back in the UK.
The 49-year-old Scot has a year remaining on his contract and met with Everton chairman Bill Kenwright at the end of the season to seek assurances that would persuade him to sign a contract extension. Five weeks on, Moyes' new contract remains unsigned.
One key factor could, however, be that Moyes does have the final say on all football matters at Goodison Park, something he would not have at Tottenham.
Meanwhile, sources close to Villas-Boas claim the former Chelsea manager is not being considered by Spurs.
Outgoing boss Redknapp has consistently rejected suggestions that Tottenham were affected by the speculation linking him with England.
Last week he urged Spurs to resolve his future, insisting that the uncertainty could cause problems in the Tottenham dressing room.
He said: "The simple situation is I've got a year left on my contract. It's up to Tottenham whether they want to extend that contract or not.
"If they don't extend it and I go into my last year, it's not an easy one when players know you've only got a year left.
"It's up to Tottenham. If they think I'm OK and I've done a decent job and deserve an extension, they'll give it to me."
But having been the longest-serving Tottenham manager since Terry Venables, whose reign ended in 1991, he has now left the club.
Redknapp spent most of his playing career with West Ham and Bournemouth and managed both clubs before taking charge of Portsmouth in 2002.
He guided the Fratton Park club into the Premier League within 14 months as First Division champions.
Redknapp left for Pompey's bitter South Coast rivals Southampton in 2004 but returned to Fratton Park in December 2005.
Bookmakers' favourites for Tottenham job
United States coach
Former Chelsea boss
Former Liverpool manager
Correct as of 13 June at 23:30 BST
Then, after dramatically saving Pompey from relegation with a brilliant end-of-season run, Redknapp steered them to a best-ever Premier League finish of ninth in 2006-07.
They improved on that by a place in 2007-08, and won the FA Cup against Cardiff, qualifying for their first journey into Uefa Cup football.
Just months later he replaced Juande Ramos at struggling Tottenham, eventually finishing eighth in the table and qualifying for the Champions League by finishing fourth the following season.
At the turn of the year they were considered potential title challengers but a run of one victory in nine matches contributed to them slipping to fourth and losing out to Chelsea for the final Champions League place.
Harry Redknapp leaves Tottenham: Steve McClaren feels 'sorry' for Spurs
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.