Justice Alito, again in dissent, said legislative subpoenas for a president's personal papers are "inherently suspicious," adding that he believed Congress would have to show significantly more need for the material. Justice Thomas in a separate dissent argued Congress had no power to issue legislative subpoenas for the president's personal documents and could proceed only if it used its impeachment powers.

House Speaker Nancy Pelosi (D., Calif.) said after the rulings that the House would continue its press in lower courts to obtain Mr. Trump's financial records as part of its oversight of the executive branch.

"The path that the Supreme Court has laid out is one that is clearly achievable by us in the lower court and we will continue to go down that path," she said.

Rep. Jamie Raskin (D., Md.) a former law professor who sits on the Judiciary Committee, called the rulings "the best we could have hoped for, given who sits on the court. It reads a lot to me like a compromise, " that drew in both conservatives and liberals.

With the election looming in November, the decisions "create a beat-the-clock moment," Mr. Raskin said. The court's framework "certainly does not make it impossible for Congress to get these records, but it does put more rungs on the ladder for Trump lawyers to hold things up."

On Twitter, former Vice President Joe Biden, the president's presumptive Democratic challenger, pointed to an October 2019 video of him calling on Mr. Trump to " release your tax returns or shut up."

The president has been fighting the release of his tax and financial documents since his presidential campaign, when he became the first presidential candidate in four decades not to release his tax returns.

During the campaign, Mr. Trump promised to release his tax returns following the completion of an audit, but he never followed through on that promise. Polls show the majority of the public believe he should release his tax returns.

The rulings give Democrats another opening to continue criticizing the president for declining to release the documents. The president's detractors have alleged that his reluctance indicates he has something to hide, an accusation the White House has denied.

Mr. Trump's niece, Mary L. Trump, wrote in a memoir set to be released next week -- a copy of which was reviewed by the Journal -- that she leaked critical financial documents about the Trump family to the New York Times in 2017 in an effort to damage the president.

In 1974, the Supreme Court required President Nixon to obey a subpoena for tapes and other records related to the Watergate investigation. In 1997, the court likewise ordered President Clinton to comply with a private lawsuit brought against him over sexual harassment allegations.

Rebecca Ballhaus and Andrew Restuccia contributed to this article.

Write to Jess Bravin at jess.bravin@wsj.com and Brent Kendall at brent.kendall@wsj.com