When ML algorithms are deployed to automate human-related decisions, human agents may learn the underlying decision policies and adapt their behavior. Strategic Classification (SC) has emerged as a framework for studying this interaction between agents and decision-makers to design more trustworthy ML systems. Prior theoretical models in SC assume that agents are perfectly or approximately rational and respond to decision policies by optimizing their utility. However, the growing prevalence of LLMs raises the possibility that real-world agents may instead rely on these tools for strategic advice. This shift prompts two questions: (i) Can LLMs generate effective and socially responsible strategies in SC settings? (ii) Can existing SC theoretical models accurately capture agent behavior when agents follow LLM-generated advice? To investigate these questions, we examine five critical SC scenarios: hiring, loan applications, school admissions, personal income, and public assistance programs. We simulate agents with diverse profiles who interact with three commercial LLMs (GPT-4o, GPT-4.1, and GPT-5), following their suggestions on effort allocations on features. We compare the resulting agent behaviors with the best responses in existing SC models. Our findings show that: (i) Even without access to the decision policy, LLMs can generate effective strategies that improve both agents' scores and qualification; (ii) At the population level, LLM-guided effort allocation strategies yield similar or even higher score improvements, qualification rates, and fairness metrics as those predicted by the SC theoretical model, suggesting that the theoretical model may still serve as a reasonable proxy for LLM-influenced behavior; and (iii) At the individual level, LLMs tend to produce more diverse and balanced effort allocations than theoretical models.