"I'm thrilled that Kenny has joined my staff as an assistant coach. He has an outstanding ability to forge relationships with players and improve their skills," Thibodeau said in the team release. "He knows what it takes to win and has learned from one of the best coaches there is in John Calipari. Kenny will be a tremendous addition to our organization."
ESPN's Adrian Wojnarowski first reported the news Tuesday.
Payne, a first-round NBA draft pick in 1989, had been on the Wildcats staff under John Calipari since 2010. He started his coaching career at the University of Oregon in 2004.
Calipari posted a statement about his assistant's departure on Twitter:
"This is a bittersweet day for us at [Kentucky basketball]. Kenny has been an extremely important part of our success here. His relationship with our players is second to none. KP coaches them and challenges them as hard as anyone, but they respect him, they listen to him and they learn from him because they know he cares. As a basketball coach and a teacher, he could run any program in the country. Simply, he's as good as they get.
"This is one of those opportunities Kenny had to look at, and I fully support his decision to join Tom Thibodeau's staff. Kenny's approach with his players, his knowledge of and his experience in the game, and his work ethic will serve him well on the professional level. I have high expectations for him in the NBA. For [Julius Randle] and [Kevin Knox], it will be immediate, but as other players get to know and trust him, they'll know he's about one thing: their improvement and their success.
"I have no doubt that Kenny is going to be a great asset to Coach Thibodeau's staff."
Payne said Saturday he'd have a difficult time leaving Kentucky after the Knicks reached out.
"But I have a great job for a lot of different reasons," he told reporters. "And all I can tell you is that I've been blessed to be able to be in a position to help a lot of people. It's more than just basketball with me when you're in a position to help families."
Knox, who played at Kentucky during the 2017-18 season, reacted to the hire:
The Knicks hired Thibodeau in July after previous stints as head coach with the Chicago Bulls and Minnesota Timberwolves. It was the first major step as the organization moves in a new direction under president Leon Rose, who arrived in March.
New York posted a 21-45 record during the 2019-20 campaign and missed the playoffs for the seventh consecutive year.