"He just said he'd like to come and play here at Tottenham," Redknapp told reporters after Spurs beat Fulham 1-0 on Saturday to move up to fourth in the Premier League.
Galaxy's Major League Soccer (MLS) season restarts on March 15. Former England captain Beckham played on loan at AC Milan during similar breaks in 2009 and 2010.
"We'd have to (move fast)," said Redknapp. "I've left it to the people at the club (Spurs). I've passed it on to them. They're talking to whoever they have to talk to at his club."
Redknapp said Darren Eales, Tottenham's director of football administration, was trying to persuade Galaxy to allow Beckham to join Spurs who are in the last 16 of the Champions League.
"We'll contact LA Galaxy to see if they will let him go and if David is up for a three-month deal, we will definitely be interested," Redknapp had told reporters on Friday.
Beckham, 35, said in October he would not go out on loan again after missing last year's World Cup in South Africa with an Achilles tendon injury picked up playing for AC Milan, who face Spurs in the Champions League first knockout round next month.
The Londoner, who has 115 England caps, has already turned down the opportunity to join Everton on loan while Galaxy could also be reluctant to let him go after last season's injury.
He has maintained his desire to play for his country and a temporary move to London could appeal as he looks to force his way back into England manager Fabio Capello's plans.
"We've got no one else to play on the right," said Redknapp. "David Bentley keeps picking up injuries on a Friday morning. It's a no brainer."
"I'm sure he (Beckham) can still do a great job at the top level otherwise I wouldn't bother.
"He isn't going to come here and get big money - we aren't massive payers at this club. But he wants to come over here to the freezing cold and play football.
"Doesn't that tell you something about the boy, that he lives in Los Angeles and might be a billionaire?," added the Spurs manager.
Redknapp's team visit Everton on Wednesday before facing United in their next home league match on January 16.
