The Premier League side have agreed to pay £3.5m for the 22-year-old, who wanted guarantees of first team football that aren’t forthcoming at Elland Road presently, with Kiko Casilla nailing down the position as Marcelo Bielsa’s preferred option in goal.
The Whites are expected to bring in a replacement for Peacock-Farrell before the transfer window shuts next Thursday at 5pm.
Burnley were able to to free up some cash by agreeing to the sale of Tom Heaton, their No 1 for the second half of last season, to Aston Villa.
Peacock-Farrell's contract at Elland Road is up next summer.
He came through the academy at Thorp Arch.
He spent time on loan with York City in the 2017/18 season and then made 29 first team appearances last season.
For Northern Ireland, he earned eight caps, including four European Championship qualifiers that all resulted in victories.