Sky Sports News understands that the five-year deal for the England international is worth an initial £40m plus add-ons, a record for an English goalkeeper.
Leeds' previous record-signing was Georginio Rutter from Hoffenheim for around £35.5m in January 2023.
Finding a new first-choice goalkeeper has been a priority for Leeds this summer after Karl Darlow's exit and an unconvincing debut season for Lucas Perri.
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Perri, who was dropped by boss Daniel Farke last season, and third-choice Alex Cairns are the only senior goalkeepers currently on Leeds' books.
Trafford takes Leeds' spending to north of £70m in this window and is expected to leave the way for Perri to join Torino on loan with an option to buy.
The 23-year-old was Leeds' top goalkeeper target heading into the summer and they hope he can bring stability to what has been a problem position for some time.
Leeds' summer business
In
Harry Wilson - Fulham, free
Tarik Muharemovic - Sassuolo, £34m
James Trafford - £40m
Out
Pascal Struijk - Brighton, £18m
Karl Darlow - Manchester United, free
Illan Meslier - Arsenal, free
Harry Gray - Sheffield Wednesday, loan
Isaac Schmidt - Young Boys, undisclosed
Jack Harrison - New England Revolution, undisclosed
