The French publication claims that Hammers and Eagles scouts were present in Bastia's 1-0 defeat to Saint Etienne, with Francois Kamano the focus of their attentions.
The 19-year-old winger, who can play on either flank and up front, has scored two goals in 20 appearances for Bastia this season but will leave the club this summer, according to the report.
Kamano moved to Bastia from Guinean side Satellite FC two summers ago, signing a three-year deal in Corsica.
But with his contract expiring in 14 months' time, the seven-cap Guinea international could be available for a cut price at the end of the campaign.
West Ham have taken Diafra Sakho and Dimitri Payet direct from French football in recent seasons, with Palace doing likewise with Pape Souare and Yohan Cabaye.
Acquiring a work permit for Kamano, whose ancestry is unclear, could prove difficult however.