The reports states that the Istanbul based club are on the lookout for a new goalkeeper following a string of high profile errors from their current keeper Tolga Zengin.
Zengin was at fault for two of the three goals Besiktas conceded against Sporting in the Europa League.
The Turkish sports newspaper Fanatik state that Ospina is close to agreeing a deal with Beşiktaş. The daily newspaper claims that initial talks between Beşiktaş and Arsenal went well but that negotiations are ongoing. Beşiktaş are also reported to have held talks with the agent of the Colombian international.
Ospina fell down the pecking order at Arsenal following the signing of Petr Cech but he was expected to get some playing time after Wojciech Szczęsny was loaned out to Roma.
The Colombian international was used in the early stages of the Champions League but was later dropped after a string of costly mistakes.