The French centre-back is out of favour at Liverpool after falling out with Klopp on a pre-season tour of the USA in 2016.
He was nominated for Palace's player of the year award last year despite only making eight appearances in the back end of the season.
West Brom's interest in the Liverpool defender comes just days after rejecting a bid from Man City for Jonny Evans.
Tony Pulis said it would be "crazy" to say they would turn down a bigger offer for the Northern Ireland international but insisted: "We're not desperate for money".
If this deal does go through it will eclipse the club's current record signing of £13m for Nacer Chadli from Spurs last summer.
It is therefore likely that the Sakho deal will be pursued further if Man City do come in with another bid for Evans.