Sakho has no future at Liverpool having fallen out of favour under Jurgen Klopp, spending the second half of last season on loan with Palace.
The Eagles are understood to have had three bids rejected for the France international, with Liverpool refusing to lower their £30m asking price.
However, the Reds would reportedly accept a bid of £25m with £5m in add-ons, and Palace were understood to be close to an agreement before West Brom entered the race at the 11th hour.
The Baggies are on the lookout for a potential replacement for Jonny Evans, who has been linked with a £30m move to either Manchester City, Leicester City or Arsenal.
Should that deal go through then West Brom would have the funds to meet Liverpool's valuation of Sakho and could provide late competition for the centre-back's signature in the final hours of the transfer window.