British media reported the deal to be worth around £40m plus bonuses, making the 23-year-old the most expensive British goalkeeper in history.
Trafford, who was included in Thomas Tuchel's England squad for this year's World Cup, made 17 appearances for City across all competitions last season.
"I know my own abilities. I go out and try as hard as I can and have as much fun as I can," Trafford said in a statement on the club website.
"I am really excited and just really happy that it is finally done. I can get onto preparing for the new season and can focus on getting in with everyone next week."
Trafford came through Manchester City's academy and gained experience during loan spells at Accrington Stanley and Bolton before joining Burnley on a permanent deal in 2023.
He returned to City in July last year but struggled to become a regular with Gianluigi Donnarumma becoming Pep Guardiola's first-choice keeper.
The English international is the third permanent addition to Daniel Farke's squad this summer, after Harry Wilson and Tarik Muharemovic.
Leeds, who finished in 14th place in the Premier League last season, will begin this year's campaign against Nottingham Forest on August 22th.