WASHINGTON — Far fewer veterans than expected are taking advantage of a new law aimed at making it easier for them to get private health care and avoid the long waits that have plagued Department of Veterans Affairs facilities nationwide.

Only 27,000 veterans have made appointments for private medical care since the VA started mailing out “Choice Cards” in November, the VA said in a report to Congress this month. The number is so small, compared to the 8.6 million cards that have been mailed out, that VA Secretary Robert McDonald wants authority to redirect some of the $10 billion Congress allocated for the program to boost care for veterans at the VA’s 970 hospitals and clinics.

Republicans and Democrats insist the problem is the department and that it needs to do a better job promoting the choice program. They also want to change a quirk in the law that makes it hard for some veterans in rural areas to prove they live at least 40 miles from a VA health site.

The government measures the distance as the crow flies, rather than by driving miles, leaving thousands of veterans ineligible.

“Veterans put their lives on the line to defend this country. The very least we can do is ensure they don’t have to jump through hoops to receive the care they need and have earned,” said Sen. Jon Tester, D-Mont., whose vast state has just one VA hospital.

The choice program was a key component of last year’s sweeping law approved in response to reports that dozens of veterans died while waiting for appointments at a VA hospital in Phoenix, and that appointment records were manipulated to hide the delays. A series of government reports said workers throughout the country falsified wait lists while supervisors looked the other way, resulting in chronic delays for veterans seeking care and bonuses for managers who falsely appeared to meet on-time goals.

The law, signed by President Barack Obama in August, allows veterans who have waited more than 30 days for an appointment to get VA-paid care from a local doctor.