How could LLMs influence our democracy? We investigate LLMs' political leanings and the potential influence of LLMs on voters by conducting multiple experiments in a U.S. presidential election context. Through a voting simulation, we first demonstrate 18 open- and closed-weight LLMs' political preference for a Democratic nominee over a Republican nominee. We show how this leaning towards the Democratic nominee becomes more pronounced in instruction-tuned models compared to their base versions by analyzing their responses to candidate-policy related questions. We further explore the potential impact of LLMs on voter choice by conducting an experiment with 935 U.S. registered voters. During the experiments, participants interacted with LLMs (Claude-3, Llama-3, and GPT-4) over five exchanges. The experiment results show a shift in voter choices towards the Democratic nominee following LLM interaction, widening the voting margin from 0.7% to 4.6%, even though LLMs were not asked to persuade users to support the Democratic nominee during the discourse. This effect is larger than many previous studies on the persuasiveness of political campaigns, which have shown minimal effects in presidential elections. Many users also expressed a desire for further political interaction with LLMs. Which aspects of LLM interactions drove these shifts in voter choice requires further study. Lastly, we explore how a safety method can make LLMs more politically neutral, while leaving some open questions.