The offer remains on the table and has not been accepted or rejected by the Baggies, Press Association Sport understands.
It tops Stoke’s offer of £17million plus £3million in add-ons from earlier in the week – which is also yet to get an official answer – but West Brom have always wanted to find a suitable replacement before allowing Berahino to complete a move.
The 22-year-old is expected to leave The Hawthorns this summer after four transfer windows of constant speculation over his future.
Last summer he threatened to strike and never play for chairman Jeremy Peace again after Tottenham had several bids rejected, the last totalling £23million but was nowhere near the basic payment Palace and Stoke have now offered.
He was omitted from boss Tony Pulis’ squad during the saga as Pulis felt the forward was not in the right frame of mind to play before being reintegrated into the squad.
Berahino scored 20 goals in 2014-15 but has not recaptured that form and Albion have accepted it is time to let the England Under-21 international, who came through their youth ranks, leave.
Palace are low on strikers after Dwight Gayle joined Newcastle and Emmanuel Adebayor left after the end of his short-term deal at the end of the season.
The Baggies are chasing West Ham‘s Diafra Sakho as a replacement for Berahino but the Hammers want reinforcements before letting him go.