The law remains unpopular with voters overall and has yet to provide a significant boost to Democrats even after the administration exceeded its enrollment goals. (JESSICA RINALDI/Reuters)

Democrats and Republicans agree on one thing about the hotly contested Affordable Care Act: When it comes to voter intensity, the GOP holds a clear upper hand.

But a trio of major liberal groups hopes to change that in coming months, with plans to spend tens of millions of dollars persuading residents in a dozen key states to vote for Democrats based on the issue. Whether they succeed could help determine not just control of the Senate but the fate of key gubernatorial races and the law’s viability.

The Service Employees International Union, the Planned Parenthood Action Fund and have each launched campaigns in recent weeks aimed at mobilizing support for the law and the officials who back it. By focusing on more popular parts of the law — including Medicaid expansion, free birth-control coverage and a bar on denying coverage for preexisting conditions — the groups hope to coax individuals who often skip voting in midterm elections to make it to the polls.

The advocacy groups say they are still honing specific plans but that the pieces are already in place. Tens of thousands of volunteers and campaign workers are already engaged, having spent the past few months dedicated to signing people up for insurance plans under state and federal health insurance exchanges.

The organizations now plan to apply that same machinery to the midterms, with a particular focus in states that did not expand Medicaid but have competitive gubernatorial or Senate races, such as Louisiana and Florida. They are shifting thousands of workers and volunteers who had been focused on enrollment to become steeped in Medicaid expansion. And they plan to return to the people they enrolled in coverage and ask them to vote.

The groups say they are devoting significant resources to the efforts. Planned Parenthood Action Fund will spend between $12 million and $15 million, according to its president, Cecile Richards. MoveOn’s PAC and other arms “will spend millions” on issue advocacy and direct support of candidates, according to executive director Anna Galland.

SEIU Healthcare executive vice president Kirk Adams, whose division has more than 1.2 million members, said the Medicaid expansion issue resonates with many voters who often sit out non-presidential election years. Polls show majorities of voters support enlarging the program with the help of federal funds, even in some deep-red states.

“We have an issue we think can motivate folks, and we feel like we have a pretty clear defining line between Candidate X and Candidate Y,” Adams said.

But they face stiff odds, in part because Democratic turnout tends to drop off significantly during midterms. And among the general electorate, GOP voters who oppose the Affordable Care Act, also known as Obamacare, are the most fired up.

A recent national Quinnipiac poll found that half of Democrats said they were more likely to vote for a candidate who supported the law, while eight in 10 GOP voters were less likely to vote for such a candidate.

“A turnout message that half of your base doesn’t care about is not very compelling,” Brad Dayspring, spokesman for the National Republican Senatorial Committee, wrote in an e-mail. “That is a 26-point intensity gap between the parties.”

The law also remains unpopular with voters overall and has yet to provide a significant boost to Democrats, even after the administration exceeded its enrollment goals.

“The real issue is whether they can make it more intense for Democrats,” said Robert J. Blendon, a health policy and political analysis professor at Harvard University. “In off-year elections, it’s intensity, not public opinion, that matters.”

Liberal groups say that the fact that 24 governors and state legislatures have declined the law’s optional expansion of Medicaid — even though the federal government would initially cover all the costs — could prove decisive in gubernatorial races in Florida, Georgia, Kansas, Michigan, Pennsylvania and Texas.

A handful of those states also have competitive Senate races, and some Democrats have already seized on the issue. During a confirmation hearing this week for the new Health and Human Services chief, Sen. Kay Hagan (D-N.C.) used her time to criticize her state’s Republican leaders for keeping about 500,000 residents off Medicaid’s rolls.

“These are some of the most vulnerable in our society, who will continue to seek care in emergency rooms and then will leave chronic conditions unmanaged, which we know is detrimental to their health and the economy,” said Hagan, who is being challenged by state House Speaker Thom Tillis (R).

In Louisiana, embattled Sen. Mary Landrieu (D) has taken aim at Gov. Bobby Jindal (R) for opposing Medicaid expansion, calling those who have been left without coverage the “Jindal gap.”

Jindal recently sued MoveOn for erecting a billboard criticizing state leaders’ decision not to expand the program. They argued that the billboard violated trademark law, but a federal judge disagreed, giving MoveOn a public-relations boost.

“We will see which races shape up to be most competitive, but we hope Obamacare, including the Medicaid blockade, will be an issue that Democrats run proudly on this fall in all of the relevant elections,” Galland said.

SEIU and two other progressive groups sent out an e-mail Friday to supporters suggesting that a 32-year-old mother of three in Florida died because she did not qualify for Medicaid coverage.

In addition to pushing Medicaid, the organizations are also trying to convince voters that they could lose other benefits under the law.

In Michigan, SEIU and the American Federation of State, County and Municipal Employees have run a television ad against GOP Senate candidate Terri Lynn Land, saying that if she’s elected, “insurance companies could go back to charging women more” and that “Land would even cut access to preventative care like mammograms.” Land said in her own ad that “I might know a little bit more about women” than her male opponent, Democrat Gary Peters.

The Planned Parenthood Action Fund will hit on similar themes, Richards said in an interview. She noted that doctors wrote 20 million prescriptions for free birth control last year because of one Obamacare provision.

“We have to actually make the connection for women, that’s why they get that benefit,” Richards said. “Women understand the difference between people who want to make their lives better and improve their health care and those who don’t.”

One of the questions these activists face is whether they can convert the infrastructure they built over the course of the initial six-month enrollment period into activities that benefit Democrats in key midterm races. Adams said SEIU will be careful to follow elections law, but since it had amassed “a very large database” as it knocked on doors to sign people up for coverage, “we have what we need to know to make it into a voter file.”

In Florida, Adams said, 2,000 activists from outside SEIU’s direct ranks are now committed to working to make Medicaid expansion a major issue in the fall campaign.

With other groups, the advocacy will be more indirect. Organizing for Action, which grew out of President Obama’s campaigns but is not working on behalf of candidates, had 1,700 “fellows” this spring — its largest class ever. All were trained on the enrollment effort. Many OFA activists go on to work for groups such as Planned Parenthood or MoveOn.

But Blendon cautioned that the dozen states most critical in the battle for the Senate — including Alaska, Arkansas, Colorado, Iowa and Kentucky — do not look like the United States more broadly. On average, he said, insurance coverage increased only 4 percent in those 12 states during the law’s initial sign-up period.

“Those states, in general, suffer from the fact that they don’t look like polls generally,” Blendon said. “They’re more Republican, the independents are more conservative, and they’re older.”