A fee in the region of £3.7m was agreed between Leeds and Everton last week for the 22-year-old, who is out of contract at Elland Road this summer, and he discussed personal terms with the Merseyside team.
But instead of joining them he has opted for West Ham because of the prospect of more regular first-team football.
The Hammers have been playing James Tomkins at right-back ahead of on-loan Carl Jenkinson and Byram would have more chances at Upton Park than at Goodison Park, where Republic of Ireland international Seamus Coleman holds the right-back spot.
Owner Massimo Cellino confirmed: "I'm really sorry for Everton because they really acted like gentlemen, but Sam chose West Ham."