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Refugees waited for the gates to open at the Horgos border control area in Serbia on Tuesday so that they could cross into Hungary.Credit Sergey Ponomarev for The New York Times

Executives from leading American aid groups, watching the muddled international response to the migrant crisis with dismay, called on the United States government on Tuesday to multiply many times over its recent commitment to take 10,000 displaced Syrians in the next year.

“I think if the United States came out and said we would take 100,000, that would change things,” said Carolyn Miles, president and chief executive of Save the Children, in a telephone news conference with colleagues from groups including Mercy Corps, Oxfam America, CARE U.S.A. and Physicians for Human Rights.

Paul O’Brien, a vice president for policy and campaigns at Oxfam, said, “This is fast becoming a legacy issue for the Obama administration, and they are not doing enough.”

Mr. O’Brien drew a numerical comparison to Lebanon, where by some calculations a quarter of the population is now made up of Syrians who have fled their country. The war, which has lasted four and a half years, has sent roughly four million Syrians into neighboring countries — and now increasingly to Europe. Half of them are younger than 18.

Continue reading the main story Graphic How Syrians Are Dying Over four years of war has forced more than four million to flee the country, fueling a migrant crisis in the Middle East and Europe.

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If the United States followed Lebanon, Mr. O’Brien said, “we could absorb the entire population of Syria three times over.”

The Obama administration is under intense pressure to play a more assertive role in resettling families displaced by Middle East mayhem, many of them from Syria. The crush has threatened to overwhelm Europe’s abilities, and governments of the European Union are engulfed in fractious disputes over how to proceed.

Last week administration officials said the United States would increase the number of Syrian refugees it accepts to 10,000 in the fiscal year beginning October from fewer than 2,000 this year. That is still a small fraction of the hundreds of thousands seeking asylum in Europe. Germany alone has said it expects 800,000 this year.

The Obama administration’s announcement caused a range of reactions in the United States, reflecting how the refugee issue has become part of a broader politically polarized debate on immigrants and entanglements in foreign conflicts.

Advocates for refugees said it was an absurdly small increase compared with the large numbers of Vietnamese and Cubans that the United States resettled in decades past.

Others have expressed concerns over inadvertently allowing Islamic extremists or others with malevolent intent to enter via a Syrian refugee surge.

The charity executives rejected those concerns, pointing to the stringent vetting required for admittance to the United States.

Andrea Koppel, vice president of global engagement and policy at Mercy Corps, said it was only a matter of time before desperate refugees fleeing turmoil in the Middle East, North Africa and Afghanistan found ways to reach the United States.

“The fact is this is a geopolitical crisis,” she said. “Our plea to the American public is: ‘Let’s not wait. Let’s open our doors.’ ”

Rather than admitting 10,000 Syrians, she said, “it should be tens upon tens of thousands.”

Donna McKay, executive director of Physicians for Human Rights, said: “The crisis we’re facing is not one in which we’re lacking information. It’s about political will and failure to act.”

Nick Osborne, vice president for international programs at CARE U.S.A., said it was “important to look beyond the numbers” of displaced people now leaving their ravaged homelands at an ever-increasing pace.

“These are people who have come from middle-class families,” he said. “These are people who could be contributors.”

Suzanne Akhras, founder and director of the Syrian Community Network, which has been helping resettle displaced Syrians in the Chicago area, also played down concerns about refugees.

“These are not people who are going to be extremists,” she said. “They want to come here for a better life.”

The executives also said they had seen huge increases in financial donations in the nearly two weeks since the photograph of Aylan Kurdi, a drowned 3-year-old Syrian boy on a Turkish beach, commandeered much of the world’s attention.

Ms. Miles, of Save the Children, said that in the first eight months of 2015, her organization had raised $200,000 for its Syria work, but that “in the last few weeks, we’ve raised $1 million.”

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