Benayoun has been without a club since being released by Chelsea at the end of last season.
As a Europa League winner and experienced Premier League and international performer, the Israel playmaker was not expected to be available for long.
But having been unable to find a new employer, Benayoun is now prepared to take in a trial spell at QPR in an effort to prove his worth.
Redknapp is happy to give the 33-year-old a chance, as squad depth is vital when piecing together a Championship promotion push, but he is waiting to see whether the former West Ham United and Liverpool midfielder arrives.
"Someone rang me about a week ago about him and I said I would be interested in having a look at him, but never heard no more," Redknapp said in the Fulham Chronicle.
"But he's very similar to what we've got here. If you get injuries, you never know, do you?"