Musk does not have "the high burden required for a preliminary injunction" to block the conversion of OpenAI, said US District Judge Yvonne Gonzalez Rogers in Oakland, California.
But Rogers wrote in Tuesday's order that she wanted to resolve the lawsuit quickly given "the public interest at stake and potential for harm if a conversion contrary to law occurred", setting the trial for the autumn.
Musk and OpenAI, which he co-founded as a non-profit in 2015 but left before it took off, have been embroiled in a year-long legal battle.
The CEO of Tesla and X social media accuses OpenAI of straying from its founding mission - to develop artificial intelligence for the good of humanity, not corporate profit.
OpenAI and its CEO Sam Altman have denied the allegations.
At stake in the lawsuit is the ChatGPT maker's transition to a for-profit model, which the start-up sees as crucial to raising more capital and competing well in the expensive AI race.
OpenAI welcomed the judge's decision on Tuesday, saying the lawsuit by Musk, who launched rival startup xAI in 2023, has "always been about competition".
OpenAI's main backer, Microsoft, did not respond to a request for comment.
Marc Toberoff, a lawyer for Musk, said they were pleased the judge "offered an expedited trial on the core claims driving this case".
"We look forward to a jury confirming that Altman accepted Musk's charitable contributions knowing full well they had to be used for the public's benefit rather than his own enrichment," Toberoff said.
The ruling comes weeks after Altman rejected a $US97.4 billion ($A155 billion) unsolicited takeover bid from a Musk-led consortium with a "no thank you".
Altman, who has said OpenAI is not for sale, alleges that Musk has been trying to slow down a competitor.
SoftBank Group is in talks to lead a funding round of up to $40 billion in OpenAI at a valuation of $US300 billion, Reuters reported in January.
That would dwarf the $US75 billion valuation xAI has discussed in a recent fundraise, according to reports.
