Redknapp had targeted signing the vastly-experienced Republic of Ireland international, who has just finished a campaign with the LA Galaxy in the US, in the new year on a short-term basis.
Keane, who counts Coventry, Leeds, Liverpool and Spurs among a long list of former clubs, is believed to be interested in a return to England on a short-term deal once again after a successful spell at Aston Villa 12 months ago, during which time he scored three goals in seven appearances.
Redknapp had hoped to sign the 32-year-old, who he knows well from his time at Spurs, when the transfer window reopens in January but now interest from Villa could hit the R's chances of landing him.
Norwich, managed by Chris Hughton who also worked with Keane at Spurs, are also being linked with a January offer for the in-demand forward.
QPR last week