The 26-year-old will join the Bundesliga side for the rest of the season, with an option to turn the transfer into a permanent deal.
Van de Beek has made just six Premier League starts for United since signing from Ajax for £35m in September 2020.
Manager Erik ten Hag said recently the Netherlands international needed a move to revive his career. I
t is understood Eintracht will pay United a loan fee and is covering the majority of his wages.
Should they want to make the deal permanent, the fee is set at 11m euros (£9.5m), with an additional 3m euros in add-ons.
Van de Beek has made 62 appearances for United across all competitions, scoring twice.
He had a loan spell at Everton in 2022 before returning to United and missed the second half of last season with a serious knee injury.
Eintracht are sixth in the German Bundesliga table, nine points off the top four, and are in the Europa Conference League play-offs.
