According to BBC Sport, the England international leaves Etihad Stadium only 12 months after returning to City from Burnley for £27m last summer. Trafford made 17 appearances for City last season, including a starring role in their Carabao Cup and FA Cup triumphs, but featured just four times in the Premier League after Gianluigi Donnarumma's arrival in September ended his hopes of becoming first choice. He was part of England's World Cup squad this summer but did not play.
The fee eclipses the previous British record of £30m, which Everton paid Sunderland for Jordan Pickford in 2017, and puts Trafford's cumulative transfer value at more than £80m across his two moves involving Manchester City.
Only Kepa Arrizabalaga (£68m) and Alisson (£62m) have commanded higher fees among goalkeepers globally, making the 23-year-old one of the most expensive shot-stoppers of all time.
As part of the deal, BBC Sport understands City have retained a 20% sell-on clause as well as matching rights, meaning they would get first refusal on any future transfer if they choose to match a rival offer.
Leeds United manager Daniel Farke has been prepared to pay a premium for Trafford despite his limited game time last season, with the West Yorkshire club needing fresh options in goal after both Karl Darlow and Illan Meslier left Elland Road this summer.
Lucas Perri and Alex Cairns remain on Leeds's books, but Perri lost his place to Darlow in the second half of last season and was frozen out of the Premier League squad following a 4-3 defeat by Newcastle on 7 January.
Leeds are banking on the potential Trafford showed largely in the Championship with Burnley, where he conceded just 16 goals in 45 appearances during their promotion-winning campaign, having previously come through Manchester City's academy and spent loan spells at Accrington Stanley and Bolton Wanderers before that move.
Trafford's signing takes Leeds's summer spending beyond £70m and follows the earlier arrivals of Harry Wilson and Tarik Muharemović as Farke looks to strengthen a squad adjusting to life back in the Premier League. Perri, meanwhile, is expected to move in the other direction, with reports suggesting Leeds are ready to sanction a loan exit for the Brazilian now they have secured their long-term No 1.