House Speaker Nancy Pelosi said Thursday she wouldn't stand in the way of the Senate moving an immigration overhaul bill before it touches climate change legislation.
Members of the Congressional Hispanic Caucus are applying pressure to President Obama and Democratic leaders to move an immigration overhaul bill soon.
"If the Senate is ready with an immigration bill, we don't want anybody holding it up for any reason," Pelosi said. "Send it to us."
But Carol Browner, Obama's top environment adviser, said earlier in the week that she thought the climate change legislation was "doable."
What remains to be seen is if Democrats can do both, especially after the intense health care overhaul has many lawmakers still reeling.
Still, Pelosi remained confident that lawmakers could move the climate change bill through the Senate. On Wednesday, the California Democrat said she looked forward to "working with the Senate" and getting a bill to the president's desk.
But Sen. Jeff Sessions, the top Republican on the Senate Judiciary Committee, asked why Congress would consider immigration overhaul with the economy still in a fragile condition.
"I am surprised by reports that the Democratic leadership plans to plow ahead with a comprehensive immigration overhaul this year," Sessions said in a written statement. "In the wake of unprecedented borrowing and spending, a bitterly partisan push to overhaul the nation's health care system, and the House's passage of an unpopular climate change bill, the American people are dubious about any talk of ramming through comprehensive reform. Doing so now would further divide the country and continue to distract the Congress from the issue of greatest concern to Americans: the economy."
Sessions noted that one in 10 Americans are unemployed in an economy in which wages are stagnant and the pace of job creation is too slow.
"In this context, there is little enthusiasm in Congress to pass legislation that would legalize millions of unlawful residents to compete with out-of-work Americans for needed jobs, further driving down pay and draining government resources," he said.
Meanwhile, Republican Reps. Darrell Issa of California and Peter King of New York sent Homeland Secretary Janet Napolitano a letter Thursday calling on the department create a comprehensive strategy to close the border to illegal immigrants, including extending the physical border fence.
"The security of our southwest border demands a strategic combination of robust virtual security and a physical border fence, in addition to sufficient staffing," the lawmakers wrote, citing a report by the Government Accountability Office that revealed the virtual network is inadequate and plagued by cost overruns and time delays.
"While the department has sidelined this program pending reassessment, it has yet to articulate an effective replacement strategy after months of review," the lawmakers wrote. "The absence of a working alternative, coupled with the administration's unwillingness to extend the physical border fence, places our homeland security at risk."
Fox News' Chad Pergram contributed to this report.