After a slow start, the influx of Syrians to the U.S. has accelerated in recent months, and annual arrivals are likely to reach 10,000 by the end of September, the amount that was promised by the Obama administration.

Last year, the U.S. pledged to resettle 85,000 refugees from all over the world in the fiscal year that began on Oct. 1, including at least 10,000 Syrian refugees.

By March...