The 34-year-old Frenchman is a free agent after his short-term contract at Juventus expired, and it is understood the Baggies have registered an interest in him.
Clarke and Anelka know each other from their time together at Chelsea where the Scot was part of the coaching set-up for the first nine months of Anelka's stay at Stamford Bridge.
Anelka has enjoyed an eventful career, which began as a trainee at Paris St Germain, from where he was snapped up for a bargain £500,000 by Arsenal in February 1997.
He spent two-and-a-half years in North London before making a £23million switch to Real Madrid, and has since had spells at PSG, Liverpool, Manchester City, Fenerbahce, Bolton, Chelsea, Shanghai Shenhua and Juve.
Anelka has 69 international caps to his name, although his involvement with France was effectively ended by an 18-game ban imposed over a row with then coach Raymond Domenech during a stormy 2010 World Cup finals campaign.
West Brom's frontline is currently in a state of flux with Romelu Lukaku having returned to parent club Chelsea after his loan spell and Shane Long having entered the final year of his contract with no extension agreed.
The club has been linked with a series of targets, including Czech Republic international Matej Vydra and Wigan's Franco Di Santo, while the name of Lille's Salomon Kalou is the latest to feature on the rumour mill.