The report claims that Orman and his team will try to negotiate a loan deal for the Colombian international.
Beşiktaş were closely linked with Ospina over the summer but the deal fell through after Arsenal loaned Wojciech Szczesny to Roma.
The Black Eagles hope to strengthen their goalkeeping position over the winter break after costly mistakes by first choice keeper Tolga Zengin.
The Colombian keeper reportedly had a verbal agreement with Arsenal boss Arsene Wenger that he would get game time in the Champions League despite the arrival of £10million keeper Petr Cech from reigning Premier League champions over the summer.
The Metro reported that Ospina could leave the north London based club over the winter transfer window if Wenger did not pick him in the Champions League.
Ospina was picked at the early stages of the competition but has since been dropped after disappointing performances in the Champions League.