WASHINGTON — From California to Rhode Island, states are confronting new concerns that their Medicaid costs will rise as a result of the federal health care law.

That’s likely to revive the debate about how federal decisions can saddle states with unanticipated expenses.

Before President Barack Obama’s law expanded Medicaid eligibility, millions of people who already were entitled to its safety-net coverage were not enrolled. Those same people are now signing up in unexpectedly high numbers, partly because of publicity about getting insured under the law.

For states red or blue, the catch is that they must use more of their own money to cover this particular group.

In California, Democratic Gov. Jerry Brown’s recent budget projected an additional $1.2 billion spending on Medi-Cal, the state’s version of Medicaid, due in part to surging numbers. State officials say about 300,000 more already-eligible Californians are expected to enroll than was estimated last fall.

“Our policy goal is to get people covered, so in that sense it’s a success,” said state legislator Richard Pan, a Democrat who heads the California State Assembly’s health committee. “We are going to have to deal with how to support the success.”

Online exchanges that offer subsidized private insurance are just one part of the health care law’s push to expand coverage. The other part is Medicaid, and it has two components.

First, the law allows states to expand Medicaid eligibility to people with incomes up to 138 percent of the federal poverty line, about $16,100 for an individual. Washington pays the entire cost for that group through 2016, gradually phasing down to a 90 percent share. About half the states have accepted the offer to expand coverage in this way.

But whether or not a state expands Medicaid, all states are on the hook for a significantly bigger share of costs when it comes to people who were Medicaid-eligible under previous law. The federal government’s share for this group averages about 60 percent nationally. In California, it’s about a 50-50 split, so for each previously eligible resident who signs up, the state has to pony up half the cost.

There could be many reasons why people didn’t sign up in the past.

They may have simply been unaware. Some may not have needed coverage. Others see a social stigma attached to the program for those with the lowest incomes. But now virtually everyone in the country is required to have coverage or risk fines. That’s more motivation to come forward.

“It’s not a bad thing that we are opening a door that should have been open before,” said Judy Solomon of the Center for Budget and Policy Priorities, which advocates for the poor.

The budget consequences are real.

“Clearly we are going to need to do our best to make sure we are working within the budget we are given,” said Deidre Gifford, Rhode Island’s Medicaid director.

States always expected that some previously eligible people would sign up, but Gifford said her state enrolled 5,000 to 6,000 more than it had projected.

In Washington state, people who were previously eligible represent about one-third of new Medicaid enrollments, roughly 165,000 out of a total of nearly 483,000. But state officials say they are treating that as a preliminary number, and the true net increase may be lower once they factor in people who drop out of the program for a host of reasons, such as getting a job with coverage.

Governors in California, Rhode Island and Washington state all strongly supported the health care law. Their outreach campaigns to promote sign-ups overall probably contributed to drawing out uninsured residents who already were entitled to Medicaid.

But researchers also are seeing increased Medicaid enrollment in states that have resisted the health care law.

A recent report from the market research firm Avalere Health found Georgia enrollment increased by nearly 6 percent. Montana saw a 10 percent rise and South Carolina 5 percent. A big exception is Texas, which has barely seen any increase.

“Anyone who didn’t budget for this is going to be behind the eight ball,” Avalere CEO Dan Mendelson said. “It’s the kind of thing governors will want to discuss with the White House.”

When the health care law was being debated in Congress, many states recognized they might face a problem if droves of already-eligible people joined Medicaid. States lobbied federal lawmakers — unsuccessfully — to get more money for that group, said Ray Scheppach, the former top staffer for the National Governors Association.

“States are concerned about this,” he said. “It’s something they had been worried about right along.”