(CNN) - Congress could kill two birds with one stone by ending the government shutdown and raising the debt ceiling all at once, a Republican congressman from New York said Wednesday.
"There is a strong possibility, if [Senate Democrats] were willing to sit down and listen to us, that we would put a package together and solve the problems at once: stop the government shutdown and deal with the debt ceiling," Rep. Michael Grimm said on CNN's "New Day."
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The federal government enters Day Two of its shutdown after the Republican-led House tried but failed to pass a piecemeal funding plan Tuesday night to pay for the District of Columbia, veterans affairs and national parks.
Even though it fell short of the required two-thirds vote, it would have had no future in the Democratic-led Senate anyway, and the White House had already promised a veto.
At issue is a strategy led by a group of about 30-40 conservative House Republicans who've been trying to chisel away at the federal health care law by tying it to the short-term funding bill, known as a continuing resolution.
“Within my own conference, obviously everyone knows, we have a far right faction that we have to deal with and we have to unify,” Rep. Grimm said. “So that’s one challenge for the Republican Party.”
But Senate Democrats aren't budging and refuse to accept any bill that both funds the government and tries to hurt Obamacare.
Also on the horizon is the October 17 deadline to raise the nation's debt ceiling or else the government goes into default. While many predict a fierce battle between Congress and the White House, Grimm said he hopes Washington can avoid dragging the country through another fiscal crisis marked by partisan standoffs.
"Let me be clear, there are some that would absolutely deal with it the same way," Grimm told CNN's Chris Cuomo. "I think there are enough of us, though, that understand we cannot fool around with the full faith and credit of the United States."
A new CNN/ORC International survey indicates that a majority of the public would point fingers at Republicans in Congress if the nation's ability to borrow more money is not increased.
Underlining a sense of urgency, the two-term congressman added: "Why not deal with it right now? Let's get everything done and finished."
Grimm is part of a small group of House Republicans that's been trying to get GOP leadership to pass a "clean" continuing resolution that funds the government but excludes anti-Obamacare provisions. They say they can work on changing Obamacare separately, a compromise some Senate Democrats have said they'd be willing to accept.
The number two House Democrat, Rep. Steny Hoyer of Maryland, said Wednesday he's been talking to members like Grimm who want to move forward with a clean bill.
"It appears that we may be getting to a place where there are gonna be enough rational Republicans to join with the Democrats and pass...a continuing resolution," Hoyer told CNN's Kate Bolduan on "New Day."
While some 800,000 so-called nonessential federal workers are being furloughed, Grimm said "there's no such thing as a 'nonessential employee'."
"We need to put our country first and take care of those that have been taking care of us," he said. "I want to say to all those people–you are essential to me. If anyone is not essential, it's the U.S. Congress."