Mr Musk walked away from his $44bn (£36bn) bid to buy Twitter earlier in July, prompting the company to sue him.
Twitter hopes that the court will order Mr Musk to complete the takeover at the agreed price of $54.20 per share.
The tech billionaire has accused Twitter of withholding information about fake accounts.
His legal team has called for the trial to be held early next year due to its complexities, but Twitter asked for a September date.
