Redknapp admitted he questioned whether his former charge at Southampton had the commitment and determination to become a successful Premier League star when he was Jones' manager.
And Jones revealed: "If I'm truthful to myself I knew I had the desire.
"I think I showed it when I went out on loan to different clubs.
"It's all in the past now and I think you can see that desire when I play every week so it's not a question to be asked."
The Trinidad & Tobago ace has welcomed comments from Sunderland boss Roy Keane that he is the best centre-forward in the Premiership.
"I was glad to hear he has that much confidence in me, the way he expresses himself is his own way," Jones told the Northern Echo. "I still have a long way to go.
"It's a huge compliment from him, as new as I am to the game it's not something I take to my head because you might lose yourself, but it's nice.
"I've had a bit of stick (from my team-mates) for that. If he says something like that about you, you might not have to live up to that, but you have to put in performances to help you to escape the b****ckings.
"The best in the league is (Emmanuel) Adebayor right now, he's the top of the goal charts. Every player has their attributes; I don't think Liverpool would have bought (Fernando) Torres if he didn't have something about him.
"At the moment he's (Adebayor) in the form right now, we're not talking about (Cristiano) Ronaldo, he's a winger. Adebayor is the in-form striker at the moment and he has the most goals so he's the top man, there's no arguments about that."