NEW YORK/WASHINGTON (Reuters) - U.S. health insurers are making their case to Republican lawmakers over how Americans sign up for individual insurance and pushing for other changes to shape the replacement of former President Barack Obama’s national healthcare law.  The health insurers, including Independence Blue Cross and Molina Healthcare Inc, are also recommending ways to put more control over insurance in the hands of states as the federal oversight of Obamacare is dismantled. They emphasize that it is crucial to keep government subsidies for low income people. These changes, described by executives, high level officials in the health insurance sector and lawmakers in nearly a dozen interviews with Reuters, include pushing for more strict enforcement of eligibility for these plans.  Because Republicans are just starting to work with the new Trump administration and the debate is fluid, it is not clear ultimately what changes will take hold. But some of these ideas have started to surface in early Republican legislation, such as a co-sponsored bill from Maine Senator Susan Collins that would keep subsidies.  The moves underscore that private insurers are quietly working on how to benefit under the Trump administration, which is focusing on deregulation in healthcare, energy and manufacturing. And they show that insurers want to save aspects of Obamacare individual plans, but cut down on the risk to their own bottom lines and any hikes in premiums that threaten the viability of this insurance market. This market for individual insurance covers about 10 million people and is small compared to the employer-based system that covers more than 160 million Americans and the government-paid programs for over 120 million people.  But it is one that insurers have described as having growth potential. While Obamacare cut the uninsured rate to 11 percent, there are still millions of uninsured Americans. The largest U.S. insurer UnitedHealth Group Inc told investors recently that it sees opportunities in new state-based markets and is talking to policymakers.  Many investors believe that the Republican deregulation push with Trump will benefit insurers. “Clearly they support the private insurers and the role that they are going to play in any sort of new market,” said Jeff Jonas, a portfolio manager at Gamco Investors in Rye, New York, which he said owns the publicly traded insurers. President Donald Trump campaigned on a promise to repeal Obama’s national healthcare law on his first day in office. He and Republicans have not presented an agreed upon replacement plan, but key issues they are expected to address include the law’s requirements for individuals to have insurance. Insurers’ main “ask” takes into account replacement plans under discussion in Congress, and largely assumes that government funds will continue to subsidize health benefits, at least for the next two to three years.  Daniel Hilferty, CEO of Independence Blue Cross in Pennsylvania, told Reuters that he advocated tightening the rules around signing up for insurance outside of the open enrollment period, and tight control of which third parties are allowed to pay premiums for patients.  Independence is part of a nationwide network of Blue Cross Blue Shield licensees such as Anthem Inc and has enrolled more than 300,000 consumers in individual plans.  Hilferty’s requests, echoed by other people in the industry who did not want to be named, are similar to demands the industry made of Obama. Enrollment outside of the regular period - and third-party groups that keep poorer, sicker patients in the private market by paying their premiums - has helped lead to hundreds of millions of dollars in losses for insurers and pushed three of the nation’s largest players out of the Obamacare market. In addition, Independence is also asking for a bigger role in signing up new customers who want to buy individual plans. Insurers sell plans both on the exchanges and off the exchanges, but subsidized plans are currently mostly sold on the government run HealthCare.gov and on state-run websites in a dozen states. “It would be really helpful if we in the industry played a more significant role in the actual enrollment process,” Hilferty said. Trump signed an executive order on Friday directing the federal government to scale back regulations, taxes and penalties related to the law. But the directive did not change the priorities outlined to Reuters by the insurers and industry sources, they said.   Insurers have built their list of top priorities assuming in part that Republicans will try to overturn the existing individual mandate, which requires Americans to pay a fee if they do not have insurance. A replacement plan would need to include some type of bonus to entice healthy people to get insurance.  That, they say, would be a step towards a good mix of sick and healthy people that will keep the plans profitable. Ideas include creating high-risk pools to keep the very sick in a separate market and offering low prices to the young and healthy.     Without a punishment for not buying insurance that is like the individual mandate, the market can’t survive, according to Dr. J. Mario Molina, Chief Executive Officer of Molina Healthcare Inc, a company that provides Medicaid for the poor and individual insurance plans on the exchanges.  “It probably needs to be a combination of both an incentive and a penalty,” Molina said.  Insurers also want to keep the cost-sharing subsidies that have made healthcare costs affordable for millions of people as well as the premium subsidies that help to reduce the monthly cost for people with low incomes. Those subsidies are part of a court case filed last year that is on hold.  “If it’s free or close to free, you are more likely to sign up in the absence of the mandate,” said Dan Mendelson, CEO of  Avalere Health, a research group and consultant that advises health insurers and is part of Inovalon Holdings. Insurers also want continued premium subsidies, skewed to keep up enrollment of younger people.  “I think if you don’t have the subsidies, then the whole thing falls apart,” said  Molina. 