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Lawmakers - who got an earful from constituents back home during August - have insisted on extra checks to make sure illegal immigrants do not have access to taxpayer-funded programs.

Senators have worked on language that would prevent illegal immigrants from buying insurance through a proposed insurance exchange envisioned in the health care reform package.

But the NCLR said that could lead to situations where some members of a family would be covered and others, including children of illegal immigrants, wouldn’t be.

Mr. Obama said legalizing illegal immigrants is a way to take the sting out of the entire issue.

But Republicans said by pushing to legalize illegal immigrants, Mr. Obama is signaling that those here illegally eventually will get access to taxpayer-funded benefits.

Still, the push to pass a legalization bill is beginning to gain steam, even as advocates fret that the White House is moving too slowly.

On Thursday, Rep. Luis V. Gutierrez, Illinois Democrat and an outspoken advocate for legalization, agreed to take leadership in writing a new, more generous bill.

“We simply cannot wait any longer for a bill that keeps our families together, protects our workers and allows a pathway to legalization for those who have earned it,” Mr. Gutierrez said. “Saying immigration is a priority for this administration or this Congress is not the same as seeing tangible action, and the longer we wait, the more every single piece of legislation we debate will be obstructed by our failure to pass comprehensive reform.”