Since the tomb and convent of Mevlana were both erected after his death there can be no question of him having lived on the site. When the Sultan'ul- Ulema, his father Bahaeddin Veled first arrived in Konya, he stayed first at the Altun-Aba (iplikci) Madrasa, and then moved with his family to take up residence in the madrasa built for him by the Seljuk Emir Bedreddin Gevhertas. After the death of Bahaeddin Veled, Mevlana remained at this madrasa, which later became known as the Molla-i Atik Madrasa, and which was demolished in later years. The site of this madrasa is thought to be the garden of the former Teacher's Training School for Girls in Konya and the road running past it.
Mevlana is also known to have taught in the Altun-Aba Madrasa, the Karatay and Kucuk Karatay (Kemaliye) Madrasas and the now demolished Atabekiye Madrasa.
Above text and pictures are from the book titled "Mevlana and the Mevlana Museum".
You can purchase "Mevlana and the Mevlana Museum" book and other Turkey related books from Explore Turkey Bookstore.