WASHINGTON—The U.S. Supreme Court allowed President Donald Trump’s administration to implement part of his temporary ban on travelers from six Muslim-majority countries and said Monday it would review lower-court rulings that found the executive actions likely unlawful.

The Supreme Court’s action is a reversal of fortune for Mr. Trump, who had been on the losing end of several lower-court decisions that blocked his March 6 executive order that sought to impose a 90-day ban on U.S. entry for people from Iran, Libya, Somalia, Sudan, Syria and Yemen. The executive order, a revised version of one issued in late January, also sought to temporarily suspend the U.S. program for admitting refugees.

The justices, in an unsigned 13-page opinion, narrowed the scope of the ban for now, ruling the president couldn’t enforce it against travelers “who have a credible claim of a bona fide relationship” with someone or some organization in the U.S. The court imposed a similar narrowing of the ban on refugees.

The justices, ruling on the court’s last day of their current term, said they would give closer consideration to the case in October, when they will hear oral arguments in the matter. By that time, the 90-day pause on travel Mr. Trump has been seeking would already be over, a fact that could moot the justices’ further consideration of the current ban.

It is possible, however, that Mr. Trump’s administration would seek to restrict travel for some foreign nationals beyond the 90 days. U.S. officials are studying vetting procedures world-wide, and it is possible that review process could result in additional countries being added to the list for restricted travel.

Travelers could see Mr. Trump’s restrictions go into effect as soon as three days.

The administration, which has said the travel ban is necessary for national security, filed an emergency appeal earlier this month at the high court asking for permission to move forward with the executive order even as litigation continues. The White House also asked the Supreme Court to review the underlying issues in the case and resolve whether the president’s ban is permissible.

Mr. Trump in a statement Monday called the high court’s action “a clear victory for our national security. It allows the travel suspension for the six terror-prone countries and the refugee suspension to become largely effective.”

“My number one responsibility as commander in chief is to keep the American people safe. Today’s ruling allows me to use an important tool for protecting our Nation’s homeland,” Mr. Trump said.

The president’s critics argued the court’s order should be viewed narrowly because many would-be travelers or refugees should be seen as having a connection to the U.S.

“This order, properly construed, should really allow for only the narrowest implementation of any part of the ban,” said Omar Jadwat, an American Civil Liberties Union lawyer who argued one of the travel ban cases in the lower courts.
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The Department of Homeland Security said it would provide details on implementation after consultation with the State and Justice Departments.

“The implementation of the executive order will be done professionally, with clear and sufficient public notice, particularly to potentially affected travelers, and in coordination with partners in the travel industry,” the department said.

Mr. Trump’s first executive order on immigration, which he signed shortly after taking office, was briefly implemented with no advanced notice and caused widespread confusion at U.S. airports. It also created anxiety in immigrant communities because its terms appeared to apply to people who already had been given permission to live in, or travel to, the U.S.

The high court’s action Monday, which partially stayed lower-court rulings against Mr. Trump, signals that the president has a fair shot of prevailing at the Supreme Court. It takes five votes on the nine-member court to grant a stay, and one of the factors the court considers is whether the party seeking the stay has a fair prospect of winning the case.

Chief Justice John Roberts, who announced the court’s action, assembled an ideologically diverse coalition for the middle-ground approach, attracting the support of maverick conservative Justice Anthony Kennedy and the court’s four liberal justices.

Three of the court’s most conservative justices— Clarence Thomas, Samuel Alito and Neil Gorsuch —offered a partial dissent that objected to the narrowing of the president’s ban. In their dissent, they wrote that they feared the compromise standard, which turns on the closeness of a person’s connections to the U.S., would be unworkable and “invite a flood of litigation until this case is finally resolved on the merits.”

States, immigrant- and civil-rights groups and individual U.S. residents have brought cases challenging Mr. Trump’s planned restrictions, which they say are less about pressing national-security threats and more about fulfilling a campaign promise to ban the entry of Muslims into the U.S.

The legal battles have set up a major court test of presidential power at the very early stages of the new administration. Making matters more unusual, Mr. Trump has tweeted his own commentary about the cases as they have progressed, including statements that he would like travel restrictions that are stronger and broader than the current ones his Justice Department has been trying to defend in court.

Judges in Hawaii and Maryland blocked the president’s plans within days of his signing of the current executive order, which had never gone into effect.  In recent weeks, two federal appeals courts also ruled against Mr. Trump on the central issues in the litigation, though each court relied upon different legal grounds in reaching its decision.

On May 25, the Fourth U.S. Circuit Court of Appeals in Richmond, Va., issued the broadest ruling, saying Mr. Trump had singled out Muslims for disfavored treatment in a way that likely violated the Constitution. The court’s decision relied heavily upon comments Mr. Trump made on the campaign trail in support of a Muslim ban, as well as other comments he has made since taking office.

The San Francisco-based Ninth Circuit took a different approach when it ruled against the president on June 12, avoiding Mr. Trump’s comments and the constitutional questions altogether. Instead, that appeals court said the president violated federal immigration law because he failed to provide sufficient justification for his ban.