Redknapp was keen on luring free agent Cole to White Hart Lane after he left Chelsea earlier this summer.
However Liverpool won the race for the England star's signature with Cole set to sign a four-year deal at Anfield once he has completed a medical.
Redknapp admits Spurs tried their hardest to sign Cole, but that the 28-year-old snubbed their overtures for football reasons rather than money.
"We tried our very hardest to get him. The chairman made Joe a fantastic offer," Redknapp wrote in his column in The Sun.
"But, in the end, we were competing with a club that has years and years of tradition and a reputation as one of the biggest clubs in the world.
"I got a lovely message from Joe to inform me of his decision.
"He explained it. I think he wants to move for football reasons. He wants to be part of a transformation at Liverpool. I think maybe getting away from London and a clean break was part of it too.
"Whatever his reasons he has been genuine the whole way through. Joe is a bit of a throw-back, if you like.
"He has class and his manners are impeccable. I will always see him as the 11-year-old boy I first clapped eyes on at West Ham and I can only wish him all the best and hope his move is successful.
"Everywhere I went this summer Spurs fans were asking me 'Are we going to get him?' Daniel Levy pushed the boat out for him, he really did.
"No one can fault the effort we made to try to get him but it wasn't to be and I fully respect Joe's decision."