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Trump is scheduled to hold a 12:30 p.m. press conference to announce the move.
Acosta served in President George W. Bush's administration as assistant attorney general for the civil rights division, He also was a member of the National Labor Relations Board.
Currently, he serves at the dean of the Florida International University College of Law.
Acosta's nomination comes a day after Trump's initial nominee, CKE Restaurants head Andrew Puzder, withdrew his nomination following a slew of criticism over past controversies and statements.
If confirmed, Acosta would be the first Latino member of Trump's Cabinet.