The 27-year-old played a key role in Mikel Arteta's side after the injury sustained by first-choice centre-back William Saliba in March, starting six Premier League matches in 2022-23, scoring one goal.
Despite ultimately missing out on the league crown to Pep Guardiola's treble-winning Manchester City, Holding was part of a Gunners side who made massive improvements under Arteta, securing Champions League football with a second-placed finish.
The centre-back made the switch from the Emirates from Bolton Wanderers in the summer of 2016, and has since featured 162 times for the North London club, winning 2 FA Cups over a seven-year period.
According to journalist David Ornstein, Arsenal have rejected a formal bid from Besiktas for Holding's services, with the offer failing to hit the Gunners' asking price for the defender.
It is understood that the Turkish Super Lig club proposed a deal worth €2.5m (£2.1m) for the former England youth international, and it is believed that Arteta's men swiftly turned down the offer.
With one year left on Holding's contract at the Emirates - plus the option to extend his terms by a further year - Arsenal are said to be in no rush to sell the centre-back, especially for as little as £2.1m.
Arteta's side have been extremely active in this transfer window but have only completed the signing of one player so far, with Kai Havertz making the move from London rivals Chelsea for £65m.
Declan Rice is very close to joining the German at the Emirates, although there is currently a delay in providing the paperwork needed to complete the England international's £105m switch across the capital.
Holding is set to face more competition for game time should he remain at Arsenal for the 2023-24 term, with the Gunners set to confirm the signing of Ajax defender Jurrien Timber.
Although the Dutchman is mostly known as a centre-back, it has been reported that Arteta intends to utilise the 22-year-old primarily at right-back - a position Ben White occupied for the majority of last term.
Besiktas are looking to fight for the Super Lig title next season after finishing third in the Turkish top flight behind Galatasaray and Fenerbahce in 2022-23.
The report also claims that Senol Gunes's men are close to agreeing a deal for former Leicester City enforcer Daniel Amartey, with the midfielder currently a free agent after his contract expired at the King Power Stadium in June.
Arsenal commenced their pre-season schedule with a 1-1 draw at Bundesliga 2 outfit Nuremberg on Thursday evening, with Holding not included in the squad for the clash at Max-Morlock Stadion.