(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.

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.

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