The American teenager is currently on a six-month loan from PSG.
The report claims the Cherries have sent their scouts over to Scotland to watch Weah.
The Hoops are reluctant to leave Weah
The Celts are keen on keeping the youngster, as his loan spell ends this season.
Weah himself has expressed his desire to stay in Scotland, as he doesn’t foresee a future in Paris, alongside the likes of Neymar and Kylian Mbappe.
Weah’s impact this season
The 19-year-old has lived up to his billing after making an impressive start since joining the Celts in January.
In fact, his heroics as a super-sub earned the Park Head side three points against St Mirren.
He also scored against the Saints in the season’s reverse fixture last week which marked his fourth goal of the season.
What Celtic will miss without Weah?
Despite having limited starts, Weah has proved himself to be one of the brightest prospects in the league.
And with Oliver Burke’s future at Park Head up in the air, Neil Lennon needs Weah to fortify his attacking options.
Interest from other clubs
Another Premier League club, Crystal Palace, are also said to be keeping an eye on the American youngster.
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Yesterday, the Bhoys were at the end of a stalemate against the Premiership’s bottom dwellers Livingston.
They will be going head-to-head with Aberdeen in the Scottish Cup semi-finals this Sunday.