The striker, who was linked with a move to Glasgow last month, is valued at around £10 million by the Cherries but the Parkhead side were keen to offset that price tag by offering a player - believed to be Olivier Ntcham - as part of their opening bid.
Ntcham has been linked with Porto and Marseille in recent weeks, and is under contract at the Hoops until summer 2022, but a loss of form coupled with injury lessened his impact this past season, and Eddie Howe's side knocked back Celtic's initial offer, according to The Sun.
A leaked Celtic transfer document suggested the club were happy to sell Ntcham this summer for the right price, but it looks as though the Hoops will have to cough up a larger cash sum if they are serious about obtaining Mousset's services.
The 23-year-old Mousset was linked with Fulham during the January transfer window, but is understood to be open to the idea swapping the English top flight for the Scottish Premiership, after viewing the career trajectories of compatriots Moussa Dembele and Odsonne Edouard.
Mousset, capped eight times by France Under-21s, joined the Cherries in 2016 for around £6 million after cutting his teeth in the game with Le Havre. A total of 14 goals in 36 games for the French side earned him a move to England, but last season he made 28 appearances scoring just one goal in the Premier League.