Lawyers for Elon Musk and Twitter discussed a range of issues during a hearing in Delaware Chancery Court over the fate of their stalled $44 billion merger agreement, including the pace and substance of the discovery process, the relevance of a whistleblower complaint and the timing of the trial.

During the first half of the hearing Tuesday, lawyers for each side took turns arguing before Chancellor Kathaleen McCormick over so-called discovery motions, or their various requests for information. They are expected later in the hearing to argue over whether Mr. Musk should be able to add allegations made by a whistleblower, Twitter’s former head of security, Peiter Zatko, to his counterclaims.

“It has been a game of hide and seek playing discovery with them,” said Andrew Rossman, one of Mr. Musk’s attorneys. “They’re not answering our basic questions. They’re taking a breath.”

Mr. Rossman added that the nonjury trial for the case should be postponed to November from October for this reason. “It’s blindingly obvious to us that the current schedule no longer works,” he said.

Twitter attorney Bradley Wilson countered that the social-media company’s team has been working around the clock to meet the defendant’s demands, which he described as “unreasonable.”

“No one is taking a breath with this case,” he said. “Twitter does not have anything to hide.”

The scope of the hearing shows how contentious this legal battle has become and marks another escalation of the dispute, which has already taken a number of dramatic twists and turns. The outcome could grant or block access for either side to more emails, texts, data and legal claims, among other things.

Twitter sued Mr. Musk in July over his attempt to walk away from their $44 billion merger agreement reached in April. Mr. Musk later filed a countersuit, accusing the company of misrepresenting the condition of its business and key metrics about the users on its platform.

On Friday, Chancellor McCormick said she would allow Mr. Musk’s lawyers to reargue their request to compel Twitter to collect and review documents from a more extensive date range. The judge denied Mr. Musk’s motion in a decision late last month.

In that ruling, she cited among other things the additional burden placed on Twitter by her decision to grant him a subset of additional information from Twitter related to spam and fake accounts, after calling a request by Mr. Musk for several years of Twitter data about such accounts “absurdly broad.”

The judge also said she would consider Mr. Musk’s motion regarding Twitter’s Slack messages, and asked both sides to be prepared to discuss whether Twitter would be willing to submit the requested messages to a modified “quick peek” arrangement. That would mean data would be provided to Mr. Musk’s attorneys for a limited duration.

Chancellor McCormick is also hearing arguments related to two pending motions from Twitter: its request for Mr. Musk’s emails via accounts at his companies, Tesla Inc. and Space Exploration Technologies Corp., and for texts between Mr. Musk and Jared Birchall, head of Mr. Musk’s family office. Twitter filed a proposed order Friday accusing Mr. Musk of “failure to engage in good faith” during the discovery process over the texts.

The most weighty request Chancellor McCormick is expected to consider is whether Mr. Musk should be allowed to use allegations made in the whistleblower complaint to bolster his case. The complaint, submitted to the Securities and Exchange Commission in July and made public last month, accuses the company of failing to protect sensitive user data and lying about its security problems.

Mr. Musk and his lawyers have said that if true, the allegations would demonstrate a breach by Twitter of certain provisions of their merger agreement. Twitter has said the complaint is “riddled with inconsistencies and inaccuracies and lacks important context.”