Besiktas look set for a busy summer of transfers as they look to assemble a squad capable of making an impact on next season's Champions League.
Giroud has fired home 98 goals in 226 appearances in all competitions since joining the Gunners from Montpellier in 2012.
Carlos Bacca and Fernando Llorente are other forwards rumoured to be on the club's wishlist for next season.
Llorente could also be offered the chance of Champions League football in a bid to tempt him away from Swansea City.
The Welsh club denied they had received an initial €3m offer for the 32-year-old, who finished last season as Swansea's top-scorer with 15 goals following his move from Sevilla.
The 34-year-old will be a free agent on 30 June and although Paris St Germain are interested in the Portuguese star, the prospect of regular football with Besiktas could be appealing.
Hart is set to leave the Etihad Stadium as he does not figure in manager Pep Guardiola's plans and with minimal interest from the Premier League the 30-year-old is likely to look abroad.
The club's current number one, Fabri, looks set to join Olympiacos and Hart would be an ideal replacement having done well on loan at Torino last season.
While Besiktas are working hard to add to their squad, they are also keen to avoid losing many of last season's title winners.
Manchester United boss Jose Mourinho expressed an interest in signing the 23-year-old, but Besiktas chief Umut Guner told Turkish-Football.com that any potential permanent transfer would have to wait another 12 months before it could be completed.