The lawsuit said Altman, along with OpenAI's co-founder Greg Brockman, originally approached Musk to make an open source, non-profit company that would develop artificial intelligence technology for the "benefit of humanity".
OpenAI, Microsoft and Musk did not immediately respond to Reuters requests for comment.
Musk co-founded OpenAI in 2015, but stepped down from the company's board in 2018. He also runs the electric vehicle maker Tesla and bought Twitter for $44 billion in October 2022.
ChatGPT, the chatbot from OpenAI, became the fastest-growing software application in the world within six months of its launch in November 2022. It also sparked the launch of rival chatbots from Microsoft, Alphabet and a bevy of startups that tapped the hype to secure billions in funding.
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Since its debut, ChatGPT has been adopted by companies for a wide range of tasks from summarizing documents to writing computer code, setting off a race amongst Big Tech companies to launch their own offerings based on generative AI.