By Caroline Humer and Susan Cornwell
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 health care 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.

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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 health care, 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.