The Hammers made an approach for Solanke in the summer transfer window and remain keen on the former Chelsea and Liverpool front-man.
But Solanke only signed a new four-year contract in September and the Cherries see him as one of their key players to keep.
Solanke, who netted twice in Bournemouth's 2-0 win over Newcastle a fortnight ago, is viewed as the focal point of the Cherries' survival hopes this season.
Bournemouth sit just four points above the drop zone and losing Solanke's goals mid-way through the campaign could be detrimental to Andoni Iraola's side's chances of avoiding the drop.
Solanke has already matched his best-ever top flight total with six goals to his name this season.
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The 26-year-old would present West Ham with additional cover for Michail Antonio who picked up a potentially serious knee injury during the international break.
He hobbled off in Jamaica's CONCACAF Nations League defeat to Canada and returned to the Hammers' Rush Green training ground having further scans.
Antonio and Danny Ings are the club's only recognised strikers but have mustered up just two goals between them this season.
Ings is yet to start a Premier League game this term while Antonio has completed a full 90 minutes on only one occasion.
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West Ham have been in the market for a new striker since offloading Gianluca Scamacca in the summer.
They have been linked with moves for Paris Saint-Germain outcast Hugo Ekitike and former Chelsea striker Timo Werner - now of RB Leipzig.