"We can hold our own with anybody," stated Redknapp.
"We've not reached [the level of] Manchester United yet, but we're getting closer. We're an improving team and we're getting up there and eventually looking to win the championship.
I couldn't say United are on another level. I can't see them going the year unbeaten
Spurs boss Harry Redknapp
"I couldn't say United are on another level. I can't see them going the year unbeaten. There are lots of teams who could beat them on any given day. Other teams have improved and got closer to them."
Manchester United manager Sir Alex Ferguson refused to talk about the sending off of his Brazilian right-back Rafael.
The 20-year-old received a second yellow card from referee Mike Dean for tripping Spurs left-back Benoit Assou-Ekotto after 75 minutes.
United striker Wayne Rooney was booked for protesting against the decision and Ferguson said: "I don't need to discuss that.
"You can see for yourself. I don't need to discuss the referee."
Rafael, who was up against Gareth Bale on the left of Tottenham's midfield, was booked in the first half for a foul on Spurs midfielder Wilson Palacios.
Tottenham's best chance came when Peter Crouch poked the ball just wide in the first half while Rafael van der Vaart went close with a header just before the break before shooting narrowly over following a late corner.
But United defenders Nemanja Vidic and Rio Ferdinand stood firm amid the home side's periods of concerted pressure.
"It was a tough game - they didn't make many chances against us," said Ferguson. "We were a bit careless with our use of the ball.
"It was a decent point for us. Our defenders were particularly good but when we were in the final third maybe we needed a better pass."
Redknapp echoed his admiration for the defensive skills on both sides.
"Both teams defended well, especially the two central defenders on their side and Michael Dawson and William Gallas on ours," said the Spurs boss. "They were fantastic, they did their jobs and made it difficult.
Van der Sar happy with point at Tottenham
"Chances were few and far between. I thought we edged the game without creating too many clear-cut chances.
"Manchester United are a Champions League winning team, they've played big games and defended for their lives in Europe. They came here and were difficult to beat.
"When they went to 10 men we couldn't have been more open. We were as open as a barn door going for it."
Redknapp also conceded that time was running out in his bid to sign David Beckham on loan from Major League Soccer side Los Angeles Galaxy.
Beckham is training with Spurs but has not been granted permission to play for the Premier League side.
Redknapp added: "David has to be back to America in February and there's not a lot of time, which is a shame."
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