His promising team are level on 12 points with leaders Chelsea after four matches but face a big test against champions Manchester United at White Hart Lane on Saturday.
Asked if Spurs were good enough to break the dominance of United, Liverpool, Chelsea and Arsenal at the top of the table, Redknapp said: "It's possible. We've got to aim to try and do that. It's time somebody broke into that top four.
"Aston Villa are looking strong, and I like the signings they have made in the (transfer) window, Everton will be pushing again and Manchester City look very, very strong. It's possible for somebody to do it so why not Tottenham?"
Asked what he would consider a successful season, Redknapp said: "A European finish would be successful, if we can get back into the Europa League, but if we could get into the Champions League it would be even better.
GREAT START
"We've had a great start and we're in good form and we've played some terrific football, as well as winning games we've scored goals and we've looked a decent team.
Tottenham, who are three points ahead of United after enjoying their best start to a season since the league and FA Cup double-winning campaign of 1960-61, have a tough trip to Chelsea in their following league match on September 20.
"I'd certainly take four points from the next two games," said Redknapp. "We'll all start believing (a top-four place) is a real possibility if we did that but it's two tough games against two of the best three teams in Europe."
Asked if United were more vulnerable after selling players like Portugal winger Cristiano Ronaldo, Redknapp said: "Any team would miss Ronaldo but the money was fantastic and whoever they play they have still got great players.
"(Wayne) Rooney's in great form, Rio (Ferdinand) looks like he'll be back alongside (Nemanja) Vidic and whenever they play together they look strong. They will be bang (up) there again at the end of the year when all the trophies are handed out."
