Pienaar wants out of his remaining three-year contract with Spurs, where he has played only two English Premier League matches this season.
However, Spurs have priced him out of the market and Redknapp believes the South African can provide cover during the second half of the season.
Redknapp said last week: "I don't have any need to weaken the squad. I would rather have a good squad for the last 18 matches of the season than let one or two go who could play a part in the run-in."
QPR made an offer believed to be below Spurs's evaluation of Pienaar, said to be at least £6 million (R72 7million), while they bought him for half that amount from Everton in January last year.
On Monday, QPR, who sacked Neil Warnock for Hughes early this month, were set to increase their bid as they are desperate to improve a team that is just one point off the relegation zone.
It is unlikely that Spurs will lower their demands if QPR's new offer does not match their price tag for the Bafana skipper.
Spurs face Wigan Athletic at White Hart Lane tonight and there's a slim chance Pienaar might have linked up with Hughes at Loftus Road by then.