France centre-back Sakho, 27, is out of favour at Anfield after falling out with Reds manager Jurgen Klopp on the 2016 pre-season tour of the USA.
He was sent on loan to Crystal Palace last season, where he made eight appearances, and was nominated for their player of the year award.
The news comes after the Baggies rejected an £18m bid from Manchester City for centre-back Jonny Evans.
Manager Tony Pulis said on Friday it would be "crazy" to say they would turn down a bigger offer for the Northern Ireland defender, but added: "We're not desperate for money."
Sakho turned down a loan move to West Brom last August after being told he can leave the club.
He has not played for the Reds since April 2016, when he was provisionally suspended after failing a drugs test before subsequently being cleared.
Sakho joined Palace on loan for the second half of last term and was not involved in Liverpool's Premier League game at Watford last Saturday or Tuesday's Champions League play-off, first-leg win over Hoffenheim.
Should a deal go through, the fee would eclipse the club record £13m West Brom paid Tottenham for Nacer Chadli 12 months ago.
The last two weeks of the transfer window can involve some cagey wheeling and dealing.
West Brom know they can sell Evans to Manchester City. If they do that, they have decent money to get in a replacement.
However, if they tell everyone that is what they intend to do, the price of their targets go up. So, they sound out clubs quietly, while telling the world Evans is going nowhere.
It may be that the Northern Irishman stays put. But Tony Pulis wants to know what the alternatives are - and having already spoken to Liverpool about Sakho once this summer, he is now talking to them again.
West Brom are interested in Liverpool's £30m-rated defender Mamadou Sakho.