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(CNN) —  

Staring at a prolonged government shutdown, Republican senators are privately planning to court Democratic senators on an immigration deal that would give President Donald Trump money for his border wall and include several measures long-sought by Democrats, according to sources familiar with the matter.

After Trump stormed out of a White House meeting with congressional leaders, GOP senators privately gathered in Sen. Lindsey Graham’s office Wednesday to discuss a way out of the logjam. The long-shot idea: propose an immigration deal that would include $5.7 billion for Trump’s border wall along with several provisions that could entice Democrats.

Those items include changes to help those who are a part of the Deferred Action for Childhood Arrivals program as well as immigrants from El Salvador and other countries impacted by the Temporary Protected Status program - along with modifications to H-2B visas.

The plan is in its very early stages. Its chances of success are still very uncertain at best, Republicans cautioned.

GOP senators pitched the idea to senior White House adviser Jared Kushner, who said if they came up with a proposal that got Trump his border wall money and could pass the Senate, the White House would be open to more discussions on the matter, the source said. He did not say Trump would endorse such a plan.

The GOP senators – who include Graham of South Carolina, Susan Collins of Maine, Thom Tillis of North Carolina, Lamar Alexander of Tennessee, Rob Portman of Ohio and Lisa Murkowksi of Alaska – discussed the plan in a hastily arranged meeting Wednesday, with several of them now planning to propose the idea to Democrats to see if there’s a enough support to break the logjam.

It’s still highly uncertain such a plan could reopen a quarter of the government that is shut down for a third week. But with bitter talks broken down between Trump and Democratic leaders, GOP senators say this may be the only option with no end in sight to the shutdown.

“I want to figure out a way so we can reopen quickly,” Murkowski said on her way to Graham’s office.

Asked if she was happy about how the Senate GOP lunch with Trump went Wednesday, she said she’d wished there had been “more clarity” on how to end this shutdown.

CNN’s Lauren Fox contributed to this report.