The report states that talks have gone well and that Cuenca met with club officials at the Özlüce training grounds.
The 24-year-old winger is expected to undergo a medical and complete his transfer later this week.
Cuenca rose through the ranks of the Barcelona youth academy before breaking into the B side in 2011 and later made his debut for the A side.
The Barcelona youth product joined Ajax on loan in 2013 and despite being part of the Eredivisie winning side he missed most of the season with a knee injury.
Cuenca joined Deportivo de La Coruna on a permanent move last summer on a one-year deal where he scored twice and provided three assists in 27 appearances in all competitions.
Bursaspor are currently managed by Ertuğrul Sağlam – the only Turkish manager to have guided a non top four Süper Lig side to the league title with Bursaspor.