Spurs had collected two points from their opening eight games when Redknapp was appointed as manager in October and his priority has always been to keep the club in the top flight.
He knows the job is not yet done but feels his team can start to look up rather than down after climbing to 11th courtesy of a 2-1 victory over Aston Villa at the weekend.
Tottenham are just six points clear of the relegation zone but they are also within sight of West Ham, who currently occupy the seventh spot that is likely to offer a route into Europe.
Redknapp said: "How many points needed to survive? I spoke to Peter Storrie at Portsmouth and he was talking about 42 points to me and that is a bit frightening. I don't see you'll need 42 points.
"When the pressure comes on it gets very difficult for some teams. It could still take 39 points, something around that mark.
"It's two wins for us. For some teams you are talking about six wins. West
Brom probably need another 16-17 points.
"We've just got to keep trying to win games. Every manager apart from the top four - plus Aston Villa, Everton and West Ham - will be looking over their shoulder."
FreefallRedknapp expects a tight finish at the bottom end of the table but would not likely to predict the final outcome.
"Every year someone goes into freefall but also every year someone pulls out of it when things look bleak," he explained.
"I was at Portsmouth two or three years ago when we looked at trying to get 20 more points with 10 games to go after I think we only had 16 or 17 all year.
"We managed it. West Ham have done something similar. Fulham did it last year.
"But I would love to finish seventh. We're playing so well in recent weeks and there's no reason we can't keep pushing up the table now."
As much as he would like Tottenham to improve next season, Redknapp does not believe it is realistic to think about challenging the top four.
"To break into the top four is very difficult. They're on another level still, in my opinion," he said.
"I certainly didn't say I see us being a top four club next year. I wouldn't put myself under that kind of pressure."