Striker Walters, 17, who is top scorer this season for United's Under-18 team, was thrilled to agree terms.
"It's one of the best feelings I've ever had," he told the club's official website.
"It couldn't have gone any better for me really. I joined the academy quite late and wasn't expecting to get a club, but Leeds came in and I got a scholarship and I thought the only thing I can do is keep kicking on and try and get a pro contract, and now it's a dream come true.
"I'd like to try and be involved with the first team. I've got to keep doing what I'm doing, keep improving and try and get more goals.
"When I look at Sam Byram, I played with him last year, he has done amazing for himself and that's what I'd like to do next season."