It is claimed the offer is £3.6 million, or €5 million, an amount the Ligue 1 side see as insufficient for the Ivory Coast international.
Rather than dismiss their bid outright, the report claims St Etienne are negotiating with Bournemouth in an attempt to find a middle ground.
Gradel signed a new deal just last Autumn taking him up until 2017. His previous contract had been set to expire this summer.
The winger has history with Bournemouth, spending time on loan at the club from Leicester City.
He enjoyed most success in England with Leeds United, who sold him in 2012 to the French side, who finished fifth in Ligue 1 last season.
Gradel played a key role in Ivory Coast winning the African Nations Cup earlier this year.
St Etienne reported on their official website last month that they expect Gradel to leave this summer, having expressed his desire for a new challenge, and the Premier League is beckoning.