Karen Tumlin, legal director of the National Immigration Law Center, said advocates for refugees and other immigrants would urge the justices this fall to lift the president’s travel ban for everyone seeking to come to the United States.

“We think it’s repugnant to our values that they might be treated differently because of where they are from or how they choose to pray,” Ms. Tumlin told reporters.

The court’s opinion sets up a historic legal clash this fall in which the justices will weigh the president’s power to set national security priorities against the need to protect individuals from discrimination based on their religious beliefs or national origin.

In saying they will take the case, the court partly endorsed the administration’s view that the president has vast powers to control who crosses the border. The court said the president’s powers to limit immigration “are undoubtedly at their peak when there is no tie between the foreign national and the United States.”

But the opinion also signaled that some of the justices might believe Mr. Trump exceeded even that broad authority when he twice sought to impose a blanket ban on entry into the United States from particular, Muslim-majority countries. With the limits imposed on Monday by the court, the government’s travel ban will be far more limited than the one he proposed in his first week in office and a later, revised ban.

For Mr. Trump, the court opinion is a rare legal victory after months in which the lower courts repeatedly chastised him for imposing a de facto ban on Muslims entering the country. In May, the United States Court of Appeals for the Fourth Circuit, in Richmond, Va., said the president’s revised order “drips with religious intolerance, animus and discrimination.”

The court’s decision could also lead to months of administrative and legal wrangling as consular officials try to determine which individuals are allowed to seek entry into the United States and which are barred by the opinion.

“We are going to be monitoring all of that,” said Becca Heller, the director of the International Refugee Assistance Project, one of the plaintiffs in the case.

The justices said the distinction should be easy to administer. “In practical terms, this means that” the executive order “may not be enforced against foreign nationals who have a credible claim of a bona fide relationship with a person or entity in the United States.”

But Justice Clarence Thomas, who issued a partial dissent on Monday that was joined by Justices Samuel A. Alito Jr. and Neil M. Gorsuch, warned that the court’s opinion would “prove unworkable” for officials at consulates around the world and would invite “a flood of litigation” from people denied entry.

“Today’s compromise will burden executive officials with the task of deciding — on peril of contempt — whether individuals from the six affected nations who wish to enter the United States have a sufficient connection to a person or entity in this country,” Justice Thomas wrote.

Based on the dissent, those three justices are likely to vote in favor of the Trump administration.

The court’s four-member liberal bloc — Justices Ruth Bader Ginsburg, Stephen G. Breyer, Sonia Sotomayor and Elena Kagan — are likely to vote against it.

That leaves the ultimate fate of the ban in the hands of Chief Justice John G. Roberts Jr. and Justice Anthony M. Kennedy.

Attorney General Jeff Sessions called the court’s decision an “important step towards restoring the separation of powers between the branches of the federal government” and he expressed confidence that the court would uphold the president’s travel ban in its entirety after it heard the case this fall.

Some opponents of the travel ban said they worried about the fate of people who might be barred from entry into the United States in the meantime.

“The Court’s ruling will leave refugees stranded in difficult and dangerous situations abroad,” said Hardy Vieux, of Human Rights First. “Many of these individuals may not have ‘bona fide relationships,’ but have strong reasons to look to the United States for protection.”

Mr. Trump’s revised executive order, issued in March, limited travel from six mostly Muslim countries for 90 days and suspended the nation’s refugee program for 120 days. The time was needed, the order said, to address gaps in the government’s screening and vetting procedures.

Two federal appeals courts had blocked critical parts of the order. The administration had asked that the lower court ruling be stayed while the case moved forward. The court granted part of that request in its unsigned opinion.

The United States Court of Appeals for the Ninth Circuit, in San Francisco, this month blocked both the limits on travel and the suspension of the refugee program. It ruled on statutory rather than constitutional grounds, saying Mr. Trump had exceeded the authority granted him by Congress.

The Supreme Court agreed to review both cases in October, noting that the government had not asked it to act faster.

The court suggested that the administration could complete its internal reviews over the summer, raising the prospect that the case could be moot by the time it was argued.

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