A loan move for the former England captain had been described by Redknapp as a "no-goer" earlier in the day, but his tone changed after Tottenham's FA Cup third-round win over Charlton.
After a week of negotiations between Tottenham and Beckham's parent club, the Los Angeles Galaxy, Redknapp initially said that an eight-week loan for the star had fallen through because of "insurance issues".
Redknapp's assertion was followed by a statement on Spurs' website, announcing that the 35-year-old would join the club - but only to train for a month as he aims to keep fit during the MLS close season.
But following Spurs' 3-0 win against their npower League One opponents, Redknapp appeared to backtrack on his previous statement, and insisted the midfielder could still join.
When asked if a deal could still happen, the Spurs boss replied: "I don't honestly know. That's being done above my head. Hopefully it could still happen. The situation is ongoing."
It was thought that the Galaxy were demanding an insurance fee for the former Manchester United man after he picked up an Achilles injury during last year's loan spell at AC Milan.
But it is understood the Galaxy are not demanding such a fee for Beckham's release, and the reason behind the impasse in the deal is because Spurs are unwilling to pay the percentage of his wages that the Americans are demanding.
"I was led to believe that insurance was a problem, but apparently that's not the case," said Redknapp, who saw his team beat Charlton thanks to a goal from Andros Townsend and a Jermain Defoe double.
Beckham will start training with Spurs at their training ground in Essex this week but the Galaxy are adamant they want their man back in time for the start of the MLS season on March 15.
Tottenham assistant manager Kevin Bond admits he is looking forward to Beckham's arrival.
"I think everyone will be looking forward to how he conducts himself, how he trains and I'm sure a lot of people will take a lot from him being here and seeing how he goes about his work," Bond said.
Should Beckham's loan deal fail to go through, Tottenham will be short of experienced right-sided midfielders if David Bentley's move to Birmingham is completed, as expected, this week.
Townsend, 19, who recently had his season-long loan cut short by Ipswich, impressed on the right on Sunday though, marking his debut with a 20-yard opening goal.
Bond was impressed and admits the youngster, who is naturally left-footed, looks to have a bright future at the club.
"I thought it was a terrific performance from someone making his debut," Bond said.
"If Harry feels he can bring someone in to improve the squad then he said he would look at it.
"(But) looking at Andros, it looks like we might benefit from having him around."
Spurs went in goalless at half-time after a disappointing first half but the home side soon sparked into life thanks to the introduction of midfielder Luka Modric.
Bond admits the Croatian gave Spurs a creative outlet they badly missed in the first period.
Bond said: "Maybe we were predictable in the first half and you need someone with that something different to unlock the door. He certainly has that."
Defoe returned to the starting line-up after a three-match ban for a red card for an elbow on James Collins during Spurs' 2-1 win at Aston Villa on Boxing Day.
Bond believes the England striker was determined to prove a point by contributing to today's win, which earned Tottenham an away tie against Fulham in the fourth round.
"I'm sure Jermain was frustrated because it was a harsh red card," Bond said.
"He was frustrated because he wasn't long being back from injury and to sit out another three games meant he was keen to get out on the pitch.
"Strikers live off goals and it's important for them, they become a different animal completely when they find the net."