A New York judge said Tuesday he will keep former President Donald Trump’s sentencing date of Sept. 18, but set a new timeline for decisions on defense motions.

New York Supreme Court Judge Juan Merchan said he will rule by next week on Mr. Trump’s attempt to boot him from the case. The defense says the judge’s donation to a liberal group and his daughter’s ties to political consultancy work should force his recusal.

Judge Merchan said he would rule on a question about Mr. Trump’s immunity from prosecution on Sept. 16, or 10 days later than initially stated.

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Mr. Trump’s lawyers say New York prosecutors ran afoul of the Supreme Court’s finding that presidents enjoy immunity for their official acts.

The defense pointed to evidence in Mr. Trump’s business-record trial in Manhattan that delved into Mr. Trump’s social media posts and activities during his first year as president in 2017.

Prosecutors say the complaint is unfounded, but Judge Merchan delayed sentencing for two months to consider the situation.

Mr. Trump, the GOP nominee for president, faces up to four years in prison on his conviction of 34 felony counts of falsifying business records.

He is unlikely to face a lengthy term, given his lack of a criminal record and the logistics of sending the Secret Service into prison with him.

Also, the judge might not want to upend the presidential campaign less than two months before the Nov. 5 election.

A Manhattan jury in May said Mr. Trump was guilty of criminally concealing a $130,000 payment to porn star Stormy Daniels.

Prosecutors said Mr. Trump made the payment through Michael Cohen and concealed it in 2017 through misidentified checks to the lawyer, signaling an intent to violate election laws.

At trial, Mr. Trump’s lawyers said the ex-president was busy running the country and thought he was paying Mr. Cohen for legal services, so his team logged the checks in that manner.

Mr. Trump also denies having a sexual encounter with Ms. Daniels.

The former porn star detailed the alleged 2006 encounter in Lake Tahoe in such detail from the witness stand that the judge objected from the bench.

Mr. Trump contended that Manhattan prosecutors could have brought charges against him years ago but waited until he launched another White House bid. He also said he couldn’t get a fair trial in the liberal stronghold of Manhattan.

A federal judge in Florida tossed out a separate criminal case alleging Mr. Trump unlawfully stored classified documents at his Mar-a-Lago estate, ruling special counsel Jack Smith was not legally appointed.

The decision, plus the Supreme Court’s immunity ruling, is forcing another federal judge in Washington to consider whether Mr. Smith can pursue a case alleging Mr. Trump conspired against the U.S. and its voters with his actions following the 2020 election.

A state case in Georgia over Mr. Trump’s post-election actions is on hold pending an appellate decision on whether Fulton County District Attorney Fani Willis should remain on the case after appointing an investigator who’d been her boyfriend. Defense attorneys said the situation posed an impermissible financial conflict for Ms. Willis.

The snags left the New York case as the only legal threat against Mr. Trump before voters have their say in November.