Spurs confirmed the 65-year-old's departure in the early hours of Thursday morning, which Redknapp - who has no intentions of retiring - felt was inevitable even if he had secured Champions League football.
Everton manager David Moyes, Wigan boss Roberto Martinez are leading candidates, along with former Chelsea head coach Andre Villas-Boas and Rafael Benitez.
Redknapp insists the current squad - which includes the likes of Gareth Bale and Luka Modric, both linked with moves away from White Hart Lane - can go on to bigger and better things, despite having faded badly over the final weeks of a campaign which had for so long promised so much.
"I leave behind some fantastic players. The only disappointment I have is that I think it was a team which could have gone on and eventually won the Premier League in the next year or two," Redknapp told reporters.
"I just wish I could have been able to see that through and be part of that because they have some fantastic players there."
Redknapp felt his depature was always on the cards, regardless of how the campaign ended - with Spurs losing out on Champions League football despite finishing fourth in the Premier League, because of Chelsea's Champions League final triumph over Bayern Munich last month.
"We missed out on third place by one point in the end, but that is how it goes," he said.
"I think the same outcome would have happened, the chairman would have gone down the same road.
"I had four great years at Spurs. All you can do is leave the club in a better state than you found it and I did that, for sure.
"I abide by their decision and don't hold grudges. That is life, we all move on."
Redknapp had transformed the fortunes of the north London club since taking charge in October 2008 and chairman Daniel Levy said in a statement the decision to remove him was one the board had not made lightly.
The former Spurs manager maintained there were no disagreements over a new deal, with suggestions he had asked for longer than the 12-month rolling terms on offer.
"It was nothing to do with contracts or me asking for anything at all. I had a year left and that was fine," he said.
"It was just a case the club decided they wanted a change and that is their decision because they own the club."
Redknapp continued: "I met with the chairman and the club have decided to move in a different direction with the manager.
"I have had four fantastic years at Tottenham and have loved every minute of it.
"It could not have gone better for me, with the football we played, finishing fourth was great.
"The fans have just been amazing for me, the support I got during this year when it was difficult for me was special, and I will never forget that."
In his first full season in charge Redknapp guided Spurs to fourth position, taking them into the Champions League for the first time.
Spurs played some exhilarating football on their way to the quarter-finals, before being knocked out by Real Madrid.
This has certainly been a dramatic campaign for Redknapp, who underwent a heart procedure in November, was cleared of tax evasion in February and then missed out on the England job in May.
Redknapp, however, insists speculation over his future and Spurs' subsequent collapse "had no effect on anything".
Nevertheless, one senior club figure told Press Association Sport: "Our form dropped dramatically after [Fabio] Capello resigned and one had to assume that his head was turned. We would like to have thought he could have delivered us a Champions League place."
Former West Ham boss Redknapp maintains he has no plans to retire, and could now be set for a move overseas, with a long-standing offer of a coaching role in Dubai said to be on the table.
"I feel great. I don't feel any different. I love football. That is what keeps me going really," he added.
"I love going into work with the players every day and that is what I will miss. I am not one for sitting at home."
Tottenham, meanwhile, have yet to confirm whether any of Redknapp's backroom staff will remain at the club, but all the coaching profiles had been removed from the official website.