The clubs have agreed a fee of £27 million plus extras that could see the deal rise to more than £30 million.
The BBC said the 25-year-old Benteke was travelling to London to have a medical and finalise personal terms.
Former Liverpool manager Brendan Rodgers signed the player from Aston Villa at the start of the 2015-16 season but the Belgian never won a regular starting place under him or his successor, German Juergen Klopp.
Benteke made a total of 14 league starts for the Anfield club and another 15 appearances from the bench, scoring nine goals.
Liverpool rejected two previous bids from Palace in an attempt to recoup most of the £32.5 million they spent to sign him from Villa, according to media reports.
Palace, who have struggled to score goals this year, travel to London rivals Tottenham Hotspur on Saturday after losing their opening game of the season 1-0 at home to West Bromwich Albion last weekend.
