Spurs lost for the first time this season last weekend to Manchester United.
Spurs face Chelsea on Sunday.
“Breaking into the Premier League top four is about as easy as breaking into Fort Knox,” said Redknapp in his Sun column.
“Although this season will see the biggest challenge yet to the established elite of Manchester United, Chelsea, Liverpool and Arsenal.
“Manchester City have a great chance and my overriding ambition is to make Tottenham part of the gang of four.
“I still have the same excitement, enthusiasm and appetite for football that I did on my first day in management with Bournemouth.
“Even though I am in my 60s, like Alex Ferguson and Arsene Wenger, there is now more of a chance than for years of a new name ousting one of the ‘old guard’ to finish in the top four.”
“My team does not play like theirs. Chelsea are a big, strong, powerful bunch. We have to go there and attack, it is our only form of defence because we do not have giants of the game like them,” he continued.
“But Spurs are getting there. I can’t look beyond the end of this season but the club is resurgent. A new stadium and training ground in the pipeline tell you that as well as performances. It’s no pipedream.
“We have beaten Liverpool this season then looked Manchester United in the eye for an hour before we got killed off.
“I have told the chairman, Daniel Levy, that it is a long-term plan, roughly 10 years, so I need a 10-year contract to go with it!
“But we have the players who can do it, they must now want it as much as I do.
“Maybe not this season, although we will give it everything.
“We’re improving and it would be historic to take Tottenham into the Champions League.
“A couple of years ago the gap between teams like mine and Chelsea widened, now I think it has closed again.
“On paper, my club still looks to have some work to do catching up with Chelsea.
“Their front three, if you include Frank Lampard, scores something like 60 goals a season. They are monstrous at set-pieces with players like Didier Drogba and although they don’t pass it like my team, they are powerful and ruthless.
“Consistency often goes out of the window in a volatile derby but long-term we must get it right week-in, week-out.”