Gignac is available on a free transfer this summer as his contract is set to expire in June, and he is yet to agree a deal to stay with Marseille.
He has been excellent for Marcelo Bielsa's side this season, currently their top scorer with 20 goals, with the club sitting third in the league.
The Sun report Tony Pulis is keen on Gignac and has had him scouted twice in recent weeks. The report claims the Baggies want to offer him a deal worth £75,000 per week, which would make him the club's highest earner.
He now has 60 goals across the last three seasons, and is currently 29-years-old.
His agent Jean-Christophe Cano was quoted by ESPN earlier this week confirming Gignac's future is still undecided, with the club unwilling even to match his current wages.