The Gunners are keen to sign the 23-year-old attacker, whose contract with the Cherries expires at the end of the season.
Mikel Arteta’s side are said to be looking at the prospect of signing Anthony on a free transfer.
He featured regularly for Bournemouth that won promotion to the Premier League last season.
The winger has scored eight goals and registered six assists last term, helping the Cherries finish just two points behind eventual Championship winners Fulham.
Arsenal could face fierce competition for Anthony’s signature, with a number of clubs across Europe believed to be interested in his signature.