Former Celtic boss Rodgers told King he was free to go out on loan and the Wales international has signed a season-long deal at Ibrox.
King said: “I've not really spoken to him about coming here specifically but I will do.
“A man who was so successful in Scottish football with the titles he won, I will call him and ask him about life in Glasgow, what Scottish football's like and stuff like that. But I have not really had the chance to do that."
King had no concerns about Rodgers sanctioning the move to Celtic's rivals.
“No, not at all," the midfielder said. “I am really grateful for him being clear with me a couple of weeks ago.
“He said it was up to me to try and find a new challenge and fortunately I have managed to do that - a massive challenge at a massive football club and one I am delighted to be at."