Redknapp's pursuit of Jones in the January transfer window angered Black Cats manager Ricky Sbragia.
Sunderland managed to keep hold of Jones and the former Southampton man signed a new four-and-a-half-year contract at the Stadium of Light.
Redknapp, who takes his Spurs side to Sunderland on Saturday, believes Sunderland were happy with the interest as it showed they had quality players at the club.
"I spoke to Niall and he said that he was quite pleased, to be honest," said Redknapp
"They probably used us as much we were interested in (Jones).
Pleased"Niall was quite pleased to go to his board and owners and say that 'Tottenham want to pay this for Jones, we don't want to sell him, but it shows that we have some real assets here'.
"He was pleased to do that - and Jones got a new contract."
In the end Tottenham re-signed Robbie Keane in the January transfer window and Redknapp is delighted to have the striker back at White Hart Lane.
"I was always interested in Robbie, but never thought there was a chance of that," added Redknapp.
"I gave the chairman a couple of strikers I was interested in, and he tried to see where there was a deal to be done with one of them.
"There wasn't a deal to be done with Kenwyne. Luckily enough, we got Robbie.
"Robbie has character.
"We talk about managers, but I'm sure when Arsene Wenger went to Arsenal and he saw people like Tony Adams in the dressing room - he knew his job was going to be very easy.
"Basically, they do it for you. We get the credit and when it doesn't go right we get the grief. But if you've got good lads in your dressing room who want to win, they can sort it out for you.
"They'll tell people to get their finger out."
BargainRedknapp also singled out Luka Modric for special praise as he has been impressed by the Croatian in recent weeks believing the midfielder is beginning to repay the £17million Spurs paid for him last summer.
"I think it was a bargain," said Redknapp.
"He could go and play anywhere. But we are looking to push Tottenham forward, and he won't want to go anywhere else.
"Alex Ferguson said to me after the game last week he changed how they played because of him. He said he took (John) O'Shea off and changed the system because Modric was causing so many problems."