Understanding travelers' route choices can help policymakers devise optimal operational and planning strategies for both normal and abnormal circumstances. However, existing choice modeling methods often rely on predefined assumptions and struggle to capture the dynamic and adaptive nature of travel behavior. Recently, Large Language Models (LLMs) have emerged as a promising alternative, demonstrating remarkable ability to replicate human-like behaviors across various fields. Despite this potential, their capacity to accurately simulate human route choice behavior in transportation contexts remains doubtful. To satisfy this curiosity, this paper investigates the potential of LLMs for route choice modeling by introducing an LLM-empowered agent, "LLMTraveler." This agent integrates an LLM as its core, equipped with a memory system that learns from past experiences and makes decisions by balancing retrieved data and personality traits. The study systematically evaluates the LLMTraveler's ability to replicate human-like decision-making through two stages of day-to-day (DTD) congestion games: (1) analyzing its route-switching behavior in single origin-destination (OD) pair scenarios, where it demonstrates patterns that align with laboratory data but cannot be fully explained by traditional models, and (2) testing its capacity to model adaptive learning behaviors in multi-OD scenarios on the Ortuzar and Willumsen (OW) network, producing results comparable to Multinomial Logit (MNL) and Reinforcement Learning (RL) models. These experiments demonstrate that the framework can partially replicate human-like decision-making in route choice while providing natural language explanations for its decisions. This capability offers valuable insights for transportation policymaking, such as simulating traveler responses to new policies or changes in the network.