Quantum Layout Synthesis (QLS) plays a crucial role in optimizing quantum circuit execution on physical quantum devices. As we enter the era where quantum computers have hundreds of qubits, we are faced with scalability issues using optimal approaches and degrading heuristic methods' performance due to the lack of global optimization. To this end, we introduce a hybrid design that obtains the much improved solution for the heuristic method utilizing the multilevel framework, which is an effective methodology to solve large-scale problems in VLSI design. In this paper, we present ML-QLS, the first multilevel quantum layout tool with a scalable refinement operation integrated with novel cost functions and clustering strategies. Our clustering provides valuable insights into generating a proper problem approximation for quantum circuits and devices. Our experimental results demonstrate that ML-QLS can scale up to problems involving hundreds of qubits and achieve a remarkable 52% performance improvement over leading heuristic QLS tools for large circuits, which underscores the effectiveness of multilevel frameworks in quantum applications.