Core-shell electrospinning is a versatile technique for tissue engineering applications due to the possibility of uploading drugs in the core layer. The drug is protected by the shell layer that also allows its release by diffusion. The aim of the present work is the development of a drug release system for the release of doxorubicin, a chemotherapeutic agent commonly used for osteosarcoma treatment. In core–shell fibres, polycaprolactone (PCL) is often proposed for the shell layer as it is bioresorbable and biocompatible, but it is hydrophobic and lacks adhesive sites for cellular attachment. In this work, a PCL-chitosan blend was used for the shell layer to increase PCL hydrophilicity and to provide cell-recognition sites to support cell adhesion. Polyvinyl alcohol (PVA) has been used for the core layer and doxorubicin was introduced with a concentration of 100 µg/mL. Core-shell fibres of dimension of 300–500 nm were successfully obtained and doxorubicin was released in a sustained way up to 28 days.