Washington (CNN) In the midst of a government shutdown caused by a budget battle over border security funding, chances of a resolution remain slim as President Donald Trump is privately telling officials and lawmakers he won't sign a bill that comes to his desk with only $1.3 billion allotted for border security, according to sources involved in the negotiations.

Democrats take over the House this week and have shown no signs of budging.

Trump, who shelved plans to travel to Florida and remained in Washington over the Christmas holiday, has continued to dig in, even as lawmakers floated potential compromises Sunday. After a two-hour lunch with Trump, Republican Sen. Lindsey Graham told reporters the President had not committed to a resolution but Graham was hopeful a compromise could be reached in the coming days.

"(The) President didn't commit but I think he's very open minded," the South Carolina legislator said after their lunch.

Graham proposed providing $5 billion for border security "in areas that make sense" in exchange for legislative changes to the Deferred Action for Childhood Arrivals program, or DACA, and immigrants with temporary protected status. Graham said Trump called the idea "interesting."

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