The duo have become United's first signings of the summer transfer window having joined the club on loan for the 2019/20 campaign.
Harrison has rejoined Marcelo Bielsa's squad having made 42 appearances in all competitions at Elland Road last term.
The 22-year-old bagged four goals and three assists during his loan spell as he helped United to a third-placed finish in the Championship.
United now also hold an option to make his move permanent next summer following his second temporary stint in West Yorkshire.
Seagulls centre-back White has also joined the club with the pair set to link up immediately with the rest of United's senior squad at Thorp Arch.
The 21-year-old is a product of the Brighton academy and has made two first-team appearances for the club.
White spent time out on loan at Newport County during the 2017/18 campaign, before joining Peterborough United in January where he made 16 appearances for the League One side.