President-elect Donald Trump’s plan to impose “extreme vetting” on foreign visitors entering the U.S. received fresh scrutiny amid the rash of violent attacks across Europe and the Middle East, with warnings that it faces logistical and political obstacles.

The extreme vetting is supposed to entail deeper background investigations of visitors from terrorist hotbeds including Iraq, Yemen and Somalia, and the use of stricter criteria for entry into the U.S., such as making belief in the supremacy of Shariah law grounds for exclusion.

A Trump transition official said the team is working on a number of border security and immigration improvements but was not ready to announce details.

The extra vetting, as well as plans to suspend acceptance of Syrian refugees into the U.S. and to secure the southern border, are key elements of Mr. Trump’s homeland security agenda aimed at stopping the Islamic State and other global jihadi groups.

Victor Asal, a political science professor at the State University of New York at Albany who specializes in national security issues, questioned the cost and effectiveness of the program.

“It would require enormous investments of resources because there are thousands and thousands of people coming into the country every day,” he said. “The question arises, ‘Is that the best use of the resources?’”

Despite the president’s constitutional authority to exclude people deemed dangerous from entering the country, he said, the program could be challenged in court if the basis of screening appeared biased or discriminatory, such as profiling Muslims or people of Middle Eastern descent.

“Deeming as dangerous is different then vetting,” Mr. Asal said.

Still, he said, the current screening process has holes. “There are holes in everything.”

The plan also will encounter resistance on Capitol Hill, where Democrats criticize Mr. Trump’s proposal to temporarily ban Muslims from entering the U.S. until a screening system is developed to weed out terrorists. In August, Mr. Trump replaced the proposal of a Muslim ban with tougher screening of immigrants, asylum seekers and foreign visitors based on their attitudes toward America, similar to the criteria for citizenship.

Asked about extreme vetting, Rep. Bennie G. Thompson of Mississippi, the ranking Democrat on the House Homeland Security Committee, vouched for the thoroughness of the current screening process.

“Banning entire groups of people from the U.S. is not an effective way to keep our nation secure,” he said. “I strongly support continued robust vetting for all visitors, immigrants and refugees to this country in a way that keeps our country safe and upholds American values.”

The plan to use extreme vetting came to the forefront with renewed fears of terrorist strikes after attack Monday in Turkey and Germany. In Turkey, the Russian ambassador was shot dead by an off-duty policeman who yelled, “Allahu akbar.” In Germany, a truck driver plowed into a Christmas bazaar and killed 12 people.

Kellyanne Conway, a senior adviser to the transition team, said the extreme vetting plan is popular and one key reason Mr. Trump was elected.

“His extreme vetting program is very attractive to many Americans because we simply don’t have one now. You have countries that harbor, train and export terrorist — most definitely radical Islamic terrorists — that are not vetted,” Ms. Conway said on Fox News.

“We just don’t know who lives among us. We don’t know whose migrating in from country to country and what their intentions are,” she said. “It’s not just peace through strength, but strong leadership around the globe, and stopping terrorism was a huge piece of his victory and will be a huge piece on Day One of his administration.”

Rep. Michael T. McCaul, Texas Republican and chairman of the House Homeland Security Committee, called this month for a security overhaul that included extreme vetting. In a speech at the conservative Heritage Foundation, he said the country was less safe after eight years under President Obama but Mr. Trump is “more clear-eyed about the danger.”

“We must begin extreme vetting of foreign travelers,” he said. “There are many ways to reach our shores, including as a tourist, student, immigrant or refugee. Terrorists have exploited every single one of these routes, so we need to ramp up security for all of them.”

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