Deep generative models applied to the generation of novel compounds in small-molecule drug design have attracted a lot of attention in recent years. To design compounds that interact with specific target proteins, we propose a Generative Pre-Trained Transformer (GPT)-inspired model for de novo target-specific molecular design. By implementing different keys and values for the multi-head attention conditional on a specified target, the proposed method can generate drug-like compounds both with and without a specific target. The results show that our approach (cMolGPT) is capable of generating SMILES strings that correspond to both drug-like and active compounds. Moreover, the compounds generated from the conditional model closely match the chemical space of real target-specific molecules and cover a significant portion of novel compounds. Thus, the proposed Conditional Generative Pre-Trained Transformer (cMolGPT) is a valuable tool for de novo molecule design and has the potential to accelerate the molecular optimization cycle time.