WASHINGTON — President Barack Obama’s health care law would let several million middle-class people get nearly free insurance meant for the poor, a twist that government number crunchers say they discovered only after the complex bill was signed.

The change would affect early retirees: A married couple could have an annual income of about $64,000 and still get Medicaid, said officials who make long-range cost estimates for the Health and Human Services Department.

After initially downplaying any concern, the Obama administration said late Tuesday it would look for a fix.

Up to 3 million more people could qualify for Medicaid in 2014 as a result of the anomaly. That’s because, in a major change from today, most of their Social Security benefits would no longer be counted as income for determining eligibility. It might be compared to allowing middle-class people to qualify for food stamps.

Medicare chief actuary Richard Foster says the situation keeps him up at night.

“I don’t generally comment on the pros or cons of policy, but that just doesn’t make sense,” Foster said during a question-and-answer session at a recent professional society meeting.

“This is a situation that got no attention at all,” Foster said. “And even now, as I raise the issue with various policymakers, people are not rushing to say … we need to do something about this.”

Administration officials said Tuesday they now see the problem. “We are concerned that, as a matter of law, some middle-income Americans may be receiving coverage through Medicaid, which is meant to serve only the neediest Americans,” said Health and Human Services spokesman Richard Sorian. “We are exploring options to address this issue.”

Administration officials and senior Democratic lawmakers initially defended the change, saying it wasn’t a loophole but the result of a well-meaning effort to simplify the rules for deciding who would get help under the new health care law. Instead of a hodgepodge, there would be one national policy.

But Sen. Orrin Hatch of Utah, the ranking Republican on the Senate Finance Committee, called the situation “unacceptable” and said he intended to look into it.

Governors have been clamoring for relief from Medicaid costs, complaining that federal rules drive up spending and limit state options. The program is now one of the top issues in budget negotiations between the White House and Congress. Republicans want to roll back federal requirements that block states from limiting eligibility.

Medicaid is a safety-net program that serves more than 50 million vulnerable Americans, from low-income children and pregnant women to Alzheimer’s patients in nursing homes. It’s designed as a federal-state partnership, with Washington paying close to 60 percent of the total cost.

Early retirees would be a new group for Medicaid. While retirees can now start collecting Social Security at age 62, they must wait another three years to get Medicare, unless they’re disabled.