President Obama’s health care law is putting new strains on some of the nation’s most hard-pressed hospitals, by cutting aid they use to pay for emergency care for illegal immigrants, which they have long been required to provide.

The federal government has been spending $20 billion annually to reimburse these hospitals — most in poor urban and rural areas — for treating more than their share of the uninsured, including illegal immigrants. The health care law will eventually cut that money in half, based on the premise that fewer people will lack insurance after the law takes effect.

But the estimated 11 million people now living illegally in the United States are not covered by the health care law. Its sponsors, seeking to sidestep the contentious debate over immigration, excluded them from the law’s benefits.

As a result, so-called safety-net hospitals said the cuts would deal a severe blow to their finances.