The foreign undergraduate head count in U.S. colleges and universities fell about 2 percent in the last school year, the first annual decline on that measure in more than a decade, according to a study.

The report from the Institute of International Education, funded by the State Department and made public Monday, suggests challenges for the United States as it seeks to maintain dominance in global higher education.

International undergraduates in U.S. schools totaled about 431,900 in the 2018-2019 academic year, down more than 10,000. That ended 12 straight years of growth of a crucial source of tuition revenue for colleges and universities. International students pay higher out-of-state rates at public universities and tend to receive far less financial aid than domestic students. They also contribute to the education of U.S. students, educators say, by helping to create a more diverse and cosmopolitan campus climate.

The study also found that international graduate enrollment was down in 2018-2019 for the second consecutive year, to about 377,900. That total is 3.3 percent lower than a recent peak in the 2016-2017 school year.

These patterns have raised questions about the effect of President Trump’s rhetoric and policies on the international market. Critics of the president say he has set an unwelcoming tone toward China and other major suppliers of students, citing tighter scrutiny of visa applications and threats of trade wars.

But the State Department noted that a decline in new student enrollment — as opposed to total enrollment — was first observed in 2016, before Trump took office. Marie Royce, assistant secretary of state for educational and cultural affairs, told reporters on a conference call this past week that the Trump administration is taking aggressive steps to promote U.S. higher education to potential students overseas.

“We are currently working on a high-quality global marketing campaign,” Royce said, to reinforce “the message that we welcome international students.”

Trump has joined in the pitch.

“We want all the people that want to come over from China,” he told reporters Oct. 11 as he was meeting with Chinese Vice Premier Liu He in the Oval Office. “We have the greatest university system in the world, and we’re going to keep it that way. And one of the reasons it’s great is we have a lot of students from China.”

China remains by far the largest supplier of international students. There were about 369,500 Chinese students in the United States in the last school year, counting those in undergraduate, graduate and non-degree programs as well as in temporary employment related to their studies. India ranked second with about 202,000 students, South Korea third (more than 52,000) and Saudi Arabia fourth (more than 37,000). But the Saudi total fell sharply in the 2018-2019 school year, contributing significantly to the overall decline in international undergraduates.

Allan E. Goodman, president of the Institute of International Education, said many foreign families worry about high tuition and the expenses associated with studying in the United States. “Everywhere I travel, talking with parents and students, the number one concern they have is about cost,” he said.

That could make universities in Europe, Asia and Australia more attractive for some potential students even though the United States has the largest collection of premier liberal arts colleges and research universities.

Perhaps more worrisome for U.S. schools: The data show that a historic surge of Chinese students into U.S. schools is leveling off.

The study found that the Chinese student total in the United States rose 1.7 percent in the 2018-2019 academic year. Six years earlier, that rate of growth exceeded 20 percent. But the rate has slowed each year since.

At Indiana University’s flagship campus in Bloomington, the number of Chinese undergraduates has declined 30 percent since 2017, to 1,585 this fall. University President Michael A. McRobbie, an Australian immigrant, said the decline has been manageable for a campus with more than 33,000 undergraduates. But he called it “extremely unfortunate” and said Chinese Hoosiers are “very passionate alumni.” In general, McRobbie said, “universities are hugely international institutions that welcome foreign students with open arms.”

The study, known as Open Doors, is the most authoritative guide to international enrollment and has been published since the 1948-1949 school year. The institute, a nonprofit organization founded in 1919, conducts the study in collaboration with the State Department’s Bureau of Educational and Cultural Affairs.

In a key finding, new international enrollment — students entering a U.S. institution for the first time — fell for a third straight year in the 2018-2019 school year, to about 269,400. The latest decline was 0.9 percent, much less steep than the 7 percent drop the previous year.

Data from a less comprehensive “snapshot survey” that the institute conducted of about 500 selected colleges this fall suggest a continued modest decline in new international enrollment.

Still, the United States remains the world’s most powerful magnet for college and university students. Overall, the study counted 1,095,299 international students in the United States in 2018-2019, a record total, up about 500 from the year before.

At first blush, the record seems to contradict the declines in international enrollment. But that total includes more than 200,000 foreigners who finished their schoolwork and remained in the country on student visas through a program that allows them to work temporarily in their field. That program, called Optional Practical Training, has grown significantly in recent years — offsetting the enrollment drops.