Ms. Algothany’s family is one in a rapidly growing influx of Syrian refugees — 99 percent of them Muslim — arriving in communities around the United States. Administration officials said on Friday that 8,000 Syrian refugees had been allowed into the United States since October, putting them on pace to surpass the goal of 10,000.

Through partnerships with the State Department and the Department of Health and Human Services, all of them receive assistance from nonprofit organizations that connect them to a local support network to help them find housing, register for health care and food assistance and enroll their children in school. The International Rescue Committee is one of nine nonprofit organizations helping refugees relocate and integrate into their new communities.

The increase reflects a quiet but intense push by the State Department and the Department of Homeland Security, with substantial prodding from the White House, to radically speed up the pace at which Syrian refugees are placed in the United States. It appears likely to further inflame the political debate about refugees that has become a central theme of the presidential campaign. Mr. Trump has claimed that “thousands upon thousands” of Muslim refugees with a terrorist mind-set have been “pouring into our country” without proper security screening.

The administration was spurred to action by a refugee crisis whose dimensions were driven home to the public in September, when newspapers published a photograph of the drowned corpse of Alan Kurdi, a Syrian toddler whose body washed ashore in Turkey after he and his family were tossed from a raft while attempting to flee to Greece.

“We had so much pressure to bring Syrians, that we hadn’t done enough for that crisis,” said Anne C. Richard, the assistant secretary of state for population, migration and refugees, who said she had initially questioned whether the target of accepting 10,000 Syrians — part of an overall goal of resettling 85,000 refugees in the United States this year — was feasible.