The United States international had been told he could leave Juventus after making 15 Serie A appearances for the club this season.
Although the move to Elland Road will initially be on loan for a fee in the region of £1.5 million, sources have said that Leeds will have the option to sign McKennie on a permanent deal for £35m at the end of the season.
The midfielder was left out of the Juventus squad for their 2-0 defeat against Monza on Sunday, with coach Massimiliano Allegri saying a deal had been reached for the player to leave.
McKennie, 24, will become the third USMNT player to move to Leeds, following Tyler Adams and Brendan Aaronson. The team also has a strong American presence off the pitch with manager Jesse Marsch and recently appointed first-team coach Chris Armas at the club.
Leeds have already added Max Wober and Georginio Rutter to their squad during the January transfer window as they attempt to pull clear of the relegation battle, with the team just one point clear of third-bottom Bournemouth.