WASHINGTON—Chief Justice John Roberts temporarily extended a Trump-era policy that bars asylum applicants from entering the U.S. to protect the American population from Covid-19, issuing a brief order on Monday that maintains the status quo while the Supreme Court considers an emergency request from Republican-led states to keep the exclusions in place.

In November, a federal judge found the policy unlawful because it denies applicants a chance to seek humanitarian protection. Title 42, the pandemic-era public-health measure allowing migrants to be quickly expelled back to Mexico after crossing the border illegally, was set to expire on Wednesday, leaving immigration officials scrambling to put in place a new legal regime that could stem the flow of unlawful migration.

Chief Justice Roberts, who oversees emergency matters from the District of Columbia, gave the Biden administration until 5 p.m. Tuesday to file its legal response. The temporary order is to remain in effect until the court decides the emergency request, led by the Republican attorneys general of Arizona and Louisiana.

The order doesn’t indicate the court’s view of the legal issues. But the court’s conservative majority has in other cases issued emergency orders that blocked Biden administration priorities, from public-health measures to combat Covid-19 to the plan to forgive student-loan debt. In the latter case, the court rejected the administration’s request to lift lower court orders suspending the plan and set the cases for argument in February.

In a statement Monday after the court’s order, the Department of Homeland Security said Title 42 would “remain in effect at this time and individuals who attempt to enter the United States unlawfully will continue to be expelled to Mexico. While this stage of the litigation proceeds, we will continue our preparations to manage the border in a safe, orderly, and humane way when the Title 42 public health order lifts.”

The Biden administration has been narrowing in on a plan that would combine Trump-era limits on asylum claims at the border with a new system for asylum seekers to apply to enter the U.S. legally, according to people familiar with the matter.

Title 42 is believed to have acted as a deterrent for some migrants seeking asylum because they could be turned back even if they asked for protection.

With that deterrent potentially ending, the administration is racing to put together a combination of policies aimed at staving off what is expected to be an unprecedented increase in illegal border crossings once the policy lifts, a prospect that could quickly overwhelm the Border Patrol, cities near the southern border and nonprofit shelters. Some border cities have already seen surges of migrants in anticipation of Title 42’s end.

Though officials are still completing specific policy elements, the overall plan they are moving toward would enact a new series of carrots and sticks to deter would-be asylum seekers from attempting to cross the border illegally and let them apply for admission into the U.S. instead.

A key deterrence measure would be an updated version of a short-lived Trump-era policy known as the transit ban. That ban would target migrants who cross the border illegally, subjecting them to quick deportation unless they can pass a tougher initial asylum screening, according to people familiar with the plans.

Migrants stand on the banks of the Rio Grande in El Paso, Texas, where city officials have recorded an average of about 2,500 crossings a day.

Photo: Paul Ratje for The Wall Street Journal

For migrants of some nationalities, the administration is planning to offer an alternative path into the U.S., allowing them to apply through an online portal for permission to fly to the U.S., where they can live on temporary humanitarian grounds and apply for asylum. The administration created such an immigration program for Venezuelan migrants in October, and it is expected to be expanded to Cubans, Haitians and Nicaraguans.

Some migrants will also be permitted to apply for asylum at legal land border crossings, though the administration will likely require them to register for an appointment in advance by filling out their information on a mobile application called CBP1, according to people familiar with the administration’s thinking. It couldn’t be determined how many migrants would be permitted to enter the U.S. using either pathway.

White House press secretary

Karine Jean-Pierre declined to comment Monday on specific policy measures the administration may put in place.

She discouraged migrants from coming to the U.S.-Mexico border in response to Title 42’s expiration. “It would be wrong to think that the border is open. It is not open,” she said.

Elements of the plan are likely to face immediate legal challenges from immigration advocates, who will argue they amount to a significant rollback of the nation’s legal obligation to provide protection to refugees.

At the same time, it isn’t clear whether the administration can enact its new border strategy in time or whether, even if it does, the strategy will effectively redirect many of the migrants expected to attempt to cross the border once Title 42 lifts.

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Already, cities along the border including El Paso, Texas, and Yuma, Ariz., have been overwhelmed with a new surge on top of already-elevated border crossings, leading local officials in both to declare states of emergency.

In Yuma, about 1,000 migrants have been arrested crossing a day, up from 700 to 900 over the past few weeks, according to federal authorities.

And in El Paso, city officials recorded an average of about 2,500 crossings a day in about a week through Saturday, according to data published by the city, leading some migrants to sleep on the streets in near- or below-freezing temperatures because bus and plane tickets out of the city were booked.

El Paso Mayor Oscar Leeser said city officials will continue preparations for a large influx of migrants, despite the temporary hold on the scheduled end of Title 42.

“We are preparing as though Title 42 will not long be in existence,” Mr. Leeser said.

The city declared an emergency on Saturday and is now working to set up mass shelter operations around the city. The mayor said the Red Cross has offered up to 10,000 cots.

Mario D’Agostino, an assistant deputy city manager for public safety, said Texas law enforcement and state National Guard troops are being sent to the region for additional help. Mr. D’Agostino said those officers and troops will focus on security efforts.

Cities around the country where migrants typically head, such as New York, Chicago or Washington, D.C., are also bracing for the policy’s end. In a statement on Sunday, New York Mayor Eric Adams said the city was expecting 1,000 additional asylum seekers to arrive in the city each week, many of them bused by the state of Texas.

New York City may need to reconstruct a tent-housing complex for migrants, officials said Monday. Zach Iscol, commissioner of the city’s Office of Emergency Management, said four buses of asylum seekers arrived in New York on Monday. “Everything is on the table now. This is an unprecedented emergency,” Mr. Iscol said.

Even though most experts believe repealing Title 42 will initially lead to a surge in illegal migration, the long-term effect is more uncertain. While Title 42 deterred some migrants from crossing the border, it motivated others to make repeat crossings because the policy essentially took away consequences for those caught. Once the policy is lifted, authorities will likely begin prosecuting migrants attempting to sneak into the country, which could drive down repeat crossings.

—Alicia A. Caldwell and Andrew Restuccia contributed to this article.

Write to Jess Bravin at Jess.Bravin@wsj.com