Ospina has been on international duty with Colombia playing an important role in his national team reaching the semi-finals.
The experienced keeper conceded just one goal in regulation time over the course of the tournament.
Ospina started in all but one game – the 3-2 defeat against Costa Rica in the final group game.
Besiktas president Fikret Orman recently that he wants Ospina but stated that the North London club must lower their asking price.
In a recent press conference Ospina stated: “Arsenal want a lot for him [Ospina] so we may have to look for alternatives.”
The Besiktas boss did confirm that he does want Ospina at the club and underlined that the club are targeting a foreign keeper.
“I do want Ospina and we are going to get a foreign keeper,” he added.
Ospina is reported to be unhappy at Arsenal after playing second fiddle to Petr Cech last season and is considering a move.
The Colombian international played 14 times for Arsenal last season keeping four clean sheets over the course of the season conceding a total of 17 goals.