Per the source, United are keen on the former Arsenal youth academy product who is currently valued as being worth £16.8 million.
Ozyakup has established himself as a key player at Besiktas since joining from Arsenal in 2012.
The 23-year-old midfielder played a vital role in the Black Eagles winning the league title last season and booking place in the Champions League.
United have been closely linked with a move for Juventus star Paul Pogba but Mourinho has also revealed that the Red Devils have a few other targets.
“I don’t confirm and I don’t deny, I cannot tell,” Mourinho said. “We established four important target profiles —not faces, profiles—and we got three of them at the very early stage and now we have 25 percent of our market to conclude until August 31 and that player [Pogba] is not the only one.
“We are calm, we know the profile of the player we want and we have three faces to fill that profile. I am really confident Mr. Woodward and the owners will get the profile of player that I think the squad needs to have a good balance.”
Could Ozyakup be one of the targets that has flown under the radar?
Besiktas want to keep hold of their star man after losing Gokhan Tore to West Ham on a one-year loan this summer.
The Istanbul giants will also play Champions League football next season and Ozyakup will almost certainly be guaranteed first team football in the competition if he were to stay.