Large language models (LLMs) are increasingly used for program verification, and yet little is known about \emph{how} they reason about program semantics during this process. In this work, we focus on abstract interpretation based-reasoning for invariant generation and introduce two novel prompting strategies that aim to elicit such reasoning from LLMs. We evaluate these strategies across several state-of-the-art LLMs on 22 programs from the SV-COMP benchmark suite widely used in software verification. We analyze both the soundness of the generated invariants and the key thematic patterns in the models' reasoning errors. This work aims to highlight new research opportunities at the intersection of LLMs and program verification for applying LLMs to verification tasks and advancing their reasoning capabilities in this application.