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Keyphrases are the essential topical phrases that summarize a document. Keyphrase generation is a long-standing NLP task for automatically generating keyphrases for a given document. While the task has been comprehensively explored in the past via various models, only a few works perform some preliminary analysis of Large Language Models (LLMs) for the task. Given the impact of LLMs in the field of NLP, it is important to conduct a more thorough examination of their potential for keyphrase generation. In this paper, we attempt to meet this demand with our research agenda. Specifically, we focus on the zero-shot capabilities of open-source instruction-tuned LLMs (Phi-3, Llama-3) and the closed-source GPT-4o for this task. We systematically investigate the effect of providing task-relevant specialized instructions in the prompt. Moreover, we design task-specific counterparts to self-consistency-style strategies for LLMs and show significant benefits from our proposals over the baselines.