The England centre half ended up joining Chelsea from Bolton Wanderers for a fee of around £8m in the winter window.
However, Redknapp has revealed that Spurs were in competition with their London rivals for Cahill's signature.
"We tried to get Gary Cahill, but Chelsea came in and paid him a lot more wages than we could, so we lost him," said Redknapp.
Instead, Spurs brought in former Blackburn Rovers defender Ryan Nelsen on a free transfer. The New Zealand international has yet to make a league start for the Lillywhites.