Redknapp watched Saturday's 3-1 defeat at Manchester United from the stands and took training for the first time on Sunday morning after being installed as Mark Hughes' replacement.
QPR have taken only four points from their opening 13 fixtures, and Redknapp has admitted there is no cash in the kitty to sign reinforcements during the January transfer window.
That points to a move for Beckham, who is preparing to leave Los Angeles Galaxy after their MLS Cup clash with Houston Dynamo next Saturday.
"I'm sure he could still play in the Premier League," Redknapp told BBC Radio Five Live's Sportsweek programme.
"He is a top player and a fantastic person.
"We had him at Tottenham training and he was amazing around the place, absolute class.
"I haven't discussed it with the chairman (Tony Fernandes) at the moment so we'd have to look at that but David could still certainly be a great asset to anybody, showing people how to train, how to look after yourself.
"His professionalism is second to none."
A need to be creative in the January transfer market should certainly play to Redknapp's strengths.
Yet the clear problem is a massive wage bill, swollen by the 11 new arrivals last summer, players Mark Hughes' deputy Mark Bowen insisted on Saturday were good enough for what lies ahead, even if they have rarely shown it.
"It has been difficult because so many new players came in at the start of the season," said Bowen.
"We firmly believe they are top players but getting to grips with the pace and the power and intensity of the Premier League has been a problem."
Redknapp has five weeks and eight games to assess how many are up to the task before he can do something about it, knowing if results do not improve during that time, Rangers can forget about preserving their top-flight status anyway.
"They brought a lot of players in in the summer so I don't see an awful lot more coming in in the window," he said.
"You can't just keep loading up the squad with more players. If things don't go right you're going to end up being in big trouble financially.
"We are really maybe looking at one or two loans in the window.
"There's a decent enough squad here but for whatever reason something's wrong.
"The manager's the one who's suffered but to have four points from 13 games is pretty horrendous and the players really have got to take responsibility."
Redknapp has earned his reputation from spells at Portsmouth and Tottenham, both of whom were in dire straits when he took over but he managed to guide them to safety.
It has not always worked though, and he was unable to keep Southampton in the top flight eight seasons ago.
And, worryingly for QPR, who after their Old Trafford defeat travel to Sunderland for Redknapp's first game in charge on Tuesday, their manager views the present situation as far more perilous than the one he inherited at Tottenham.
"We're deeper in," he said.
"We're 13 games gone rather than eight and, let's be honest, Tottenham should not have ended up in that position at that stage because they had a fantastic squad.
"We've got to get cracking quickly.
"The target is fourth from bottom. We've just got to get out of trouble. Four points from 13 makes it very difficult but we've got to do it."
Redknapp re-iterated that QPR's owners do not have a "bottomless pit" of money for new signings in January, stressing that he also does not have "a magic wand" to wave over the current squad.
But that does not mean the 65-year-old is not ambitious despite the position his new side are in.
"I'm not here for a short-term fix, keep the club up...I want to build something here," Redknapp told QPR Player.
"I want to stay up and then build a team next year that can progress and eventually be up in the top half of the table every season.
"I thought the lads worked hard yesterday. For 60 minutes we were right in the game and went 1-0 up. After (Armand) Traore came off we became a bit disorganised unfortunately with Anton (Ferdinand) having to play left-back.
"We got done on a couple of set plays but up until then we matched Man United all the way and deserved to be where we were and could have gone two up.
"There were some good signs there but there has to be that 90-minute consistency to pick up some results because to be honest you don't have four points from 13 games...there has to be something fundamentally wrong.
"Unfortunately for Mark it cost him his job but it's not just Mark or the coaching staff, it comes down to the players at the end of the day. They are the ones who got us into the mess we are in and have to get us out."
Asked if he was facing the biggest challenge of his managerial career, Redknapp added: "I would think yes. Four points is a difficult situation but we can do it.
"A couple of wins and it can all change."