It is understood the bid was around £20million for the attacking duo – and Howe says it did come from Slaven Bilic’s side.
West Ham’s season finished at Stoke on Sunday, but Bournemouth had one game to play after their final match was abandoned due to a bomb scare at Old Trafford.
Confirming the offers for winger Ritchie, 26, and striker Wilson, 24, Howe said: “We had a bid yesterday evening.
“The disappointing thing from my perspective is why the bid has come in a day before a game – our last game of the season.
“You have all summer to bid for players, I don’t understand that.
“Two players in the squad who are due to play, [with] a lot of speculation around their futures, it was bizarre from this club’s perspective.
“The bid was rejected and we have no incentive to sell.”