In early 2013, Janet Napolitano, then the secretary of Homeland Security, testified before Congress that the agency planned to issue a report on overstay rates by December 2013. The agency did not follow through because officials said they did not have confidence in the quality of the data. Mr. Bersin said last month that the report would be issued in the next six months.

Many members of Congress and some law enforcement officials worry that terrorists could exploit the visa program because the United States does not routinely collect biometric information — fingerprints, iris scans and photographs that can be used for facial recognition — of people leaving the country. Nearly three dozen countries, including many in Europe, Asia and Africa, collect such information.

“U.S. airports and other entry and exit points were never designed with departure control in mind,” said Theresa Cardinal Brown, the director of Immigration Policy at the Bipartisan Policy Center in Washington and a Department of Homeland Security official under President George W. Bush. “If we want to do that it’s going to mean building a lot more infrastructure.”

The 9/11 Commission report called the establishment of an entry and exit biometric system “fundamental to intercepting terrorists” trying to enter the United States because it would allow law enforcement officials to determine if a traveler had overstayed a visa.

Still, efforts to build such a system to collect the information have stalled for decades. In 2004, lawmakers passed legislation that required Homeland Security officials to accelerate efforts to create an automated biometric entry and exit data system. Congress repeated its demand for a biometric exit system in 2007 and set a deadline for 2009. But the deadline passed, with the department putting into place only a handful of pilot programs.

Since then, the department has continued to struggle to meet this requirement. A 2013 report by the Government Accountability Office said the Department of Homeland Security had more than one million “unmatched” arrival records, meaning that those records could not be checked against other information showing that the individuals had left the country, but again the department could not offer a precise number.

Despite the call by some lawmakers for an exit system, airports and the airline industry have balked because it would cost airlines $3 billion, according to a 2013 Homeland Security estimate. The department issued regulations in 2008 requiring airports to collect biometric exit information, but carriers have largely ignored the regulation, and there have been no sanctions.