The Wikipedia has about this building: “The Monastery of Stoudios, more fully Monastery of Saint John the Forerunner "at Stoudios" (Greek Μονή του Αγίου Ιωάννη του Προδρόμου «εν τοις Στουδίου» Monē tou Hagiou Iōannē tou Prodromou "en tois Stoudiou"), often shortened to Stoudios, Studion, or Stoudion, (Latin: Studium), was historically the most important monastery of Constantinople (modern-day Istanbul), the capital of the Byzantine Empire. The residents of the monastery were referred to as Stoudites (or Studites). Although the monastery has been derelict for half a millennium, the laws and customs of the Stoudion were taken as models by the monks of Mount Athos and of many other monasteries of the Orthodox world; even today they have influence.
The ruins of the monastery are situated not far from the Propontis (Marmara Sea) in the section of the city called Psamathia, today's Koca Mustafa Paşa. It was founded in 462 by the consul Stoudios (Latin: Studius), a Roman patrician who had settled in Constantinople, and was consecrated to Saint John the Baptist. Its first monks came from the monastery of Acoemetae.“

Later in the article it is indicated restoration should be taking place in 2014. When I visited there was no trace of that, a lone dog seemed to be guarding the terrain, which was closed for visitors.