The Delaware judge steering the lawsuit over Twitter Inc. and Elon Musk’s stalled $44 billion takeover said the billionaire chief executive of Tesla Inc. can amend his countersuit to incorporate allegations from a whistleblower complaint by the social-media company’s former head of security.

The judge denied Mr. Musk’s request to push back a trial over the deal in Delaware Chancery Court to November from Oct. 17.

In a decision Wednesday, the judge cited a Chancery Court rule in which leave to amend should be “freely given when justice so requires.”

Chancellor Kathaleen McCormick made the decision after an hourslong hearing Tuesday covering several pending legal issues in the case. Twitter and Mr. Musk closed the hearing with oral arguments regarding their position on whether to include allegations by the whistleblower, Peiter Zatko, who was fired by Twitter earlier this year.

“I am reticent to say more concerning the merits of the counterclaims at this posture before they have been fully litigated,” Chancellor McCormick said in her ruling. “The world will have to wait for the post-trial decision.”

Shares of Twitter closed at $41.20 Wednesday, up 6.6%.

Chancellor McCormick said she would put parameters around additional requests for information related to the new claims and maintain the existing trial schedule.

“I am convinced that even four weeks’ delay would risk further harm to Twitter too great to justify,” she said.

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In the whistleblower complaint, submitted to the Securities and Exchange Commission in July and made public last month, Mr. Zatko accuses the company of failing to protect sensitive user data and lying about its security problems.

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

After the whistleblower complaint was made public, Mr. Musk’s team requested permission from the Delaware court to include those allegations in its countersuit.

In response to Chancellor McCormick’s decision Wednesday, Alex Spiro,

a lawyer for Mr. Musk, said his team is “hopeful that winning the motion to amend takes us one step closer to the truth coming out in that courtroom.” He declined to comment on the judge’s decision not to push back the trial.

A spokesman for Twitter said, “We look forward to presenting our case in court beginning on October 17th and intend to close the transaction on the price and terms agreed upon with Mr. Musk.”

The judge filed several separate decisions Wednesday on other pending motions that were the subject of the omnibus hearing, including not allowing Mr. Musk’s team to re-argue a request that Twitter produce documents from an extended timeline.

Mr. Musk’s lawyers suggested that the judge’s earlier order on the extended timeline failed to explain the basis for the court’s decision.

“That is not true,” she wrote Wednesday. “The order identifies four specific grounds” for denying the request: the timing of the request, the large volume of documents produced by Twitter already, the burden from a previous ruling asking Twitter to produce a subset of additional documents and the burden an extended timeline would place on Twitter.

She said the defendants were wrong in arguing that she “misapprehended or overlooked facts” concerning the matter and while they disagree with her decision, “disagreement is not a basis for reargument.”

Chancellor McCormick also ordered Mr. Musk to produce phone records concerning texts between himself and Jared Birchall, head of the billionaire’s family office, over a certain period.

The judge took some shots at Mr. Musk’s legal team in the ruling, reiterating frustration she expressed in the hearing.

She said their approach to answering interrogatories—legal questions—“left much to be desired” and that despite Mr. Musk’s side having less to do in producing documents ahead of trial, they “still fell short in their obligations.”

“They did a bad job of it,” she wrote in the ruling. That echoed some statements she made during oral arguments, when she said she was bothered by Mr. Musk’s lawyers’ delayed and incomplete responses to one of her orders.

In a decision resolving a fight between Twitter and Mr. Musk over producing company Slack messages from dozens of employees, the judge ruled that Twitter needed to provide ones from just two additional people: Egon Durban, a member of Twitter’s board, and Vijaya Gadde, head of legal, policy and trust at the company.

During Tuesday’s hearing, Mr. Spiro pointed to Mr. Zatko’s expertise and reputation in the security industry and chastised Twitter for not disclosing what it knew about his accusations of “egregious deficiencies” related to privacy and digital security.

He said that the whistleblower claims could support Mr. Musk’s allegations that Twitter committed fraud by misrepresenting the condition of its business and key metrics about the users on its platform.

Twitter’s team countered by describing Mr. Zatko as a disgruntled former employee with an ax to grind and saying that his unsubstantiated complaints after a notably unsuccessful tenure shouldn’t be included in the Delaware suit. Twitter’s lawyer also said Mr. Zatko’s work at the company wasn’t related to the alleged undercounting of spam and bot accounts that Mr. Musk cited in his counterclaims.

Write to Sarah E. Needleman at sarah.needleman@wsj.com