Recent reasoning-based large language models have shown strong performance on tasks with verifiable outcomes, but their use in de novo molecular generation remains limited by the lack of training environments where rewards can be computed without reference molecules. We introduce MolRGen, a benchmark and molecular verifier for training and evaluating reasoning LLMs on de novo molecular generation. MolRGen contains approximately 4,500 protein-pocket targets, resulting in 50k multi-objective optimization prompts combining docking scores with molecular properties such as QED, synthetic accessibility, logP, and physicochemical descriptors. Unlike caption-based generation or molecule-editing benchmarks, MolRGen evaluates molecules proposed from scratch by computing rewards at generation time. We benchmark general-purpose and chemistry-specialized open-source LLMs and introduce a diversity-aware top-k metric to measure whether models can generate a diverse set of high-scoring molecules. Finally, we use the verifier to fine-tune a 128B LLM with GRPO, showing improved performance, at the cost of a diversity-exploitation trade-off. MolRGen provides a scalable testbed for studying verifier-based reasoning and reinforcement learning in molecular design.