The Baggies are targeting a forward this month and the Belgian's proven Premier League goalscoring record appeals to manager Sam Allardyce.
Sky Sports News reports that Palace are prepared to listen to offers for the 30-year-old, following the arrival of French striker Jean-Philippe Mateta from Mainz, but it is understood that there is still some distance between the two clubs at present.
West Brom's finances are believed to be tight and negotiations with the Eagles have focused on how much of his current wages would be paid by each club.
The Baggies, who have already added Robert Snodgrass in this window, are hopeful that a deal can be agreed for the former Liverpool striker before the 11pm deadline on February 1.
Benteke has made 13 Premier League appearances for Roy Hodgson's side this season, scoring three times, but now seems to be surplus to requirements with both Wilfried Zaha and Jordan Ayew preferred ahead of him.body check tags ::