The New York Times, meanwhile, reported that Senate Majority Leader Mitch McConnell (R-Ky.) met with Schumer today and proposed setting aside $1.6 billion for non-wall-related border security, in addition to $1 billion that Trump could spend however he likes on immigration.
Funding for the wall - one of the president's key campaign promises - is now the source of disagreement between Republicans and Democrats in their efforts to avert a government shutdown later this week.
After pushing the deadline back for funding portions of the government by two weeks - the original deadline was December 7 - Congress and the White House have fewer than 96 hours to reach an agreement before the clock struck midnight on Friday and vast sections of the government run out of funding. McConnell called a short-term stopgap into next year "the least preferable outcome", but added, "I think a government shutdown is not a good idea".
But his latest demand for $1 billion in immigration enforcement money to spend as he sees fit was a non-starter for Senate Minority Leader Chuck Schumer (D-NY) and House Minority Leader Nancy Pelosi (D-CA) as well.
White House senior adviser Stephen Miller said Sunday that building the wall remained a top priority and that Trump was "absolutely" prepared to shut down government to achieve that goal.
"We have other ways that we can get to that $5 billion (for a border wall)", Sanders said Tuesday morning during an interview with Fox News.
Press secretary Sarah Huckabee Sanders said the White House was willing to work with Congress to avert the partial shutdown, with just days before it is due to kick in, at midnight on Friday.
Newt Gingrich & Matthew Whitaker Now Being Considered for Trump's Chief of Staff
Those angling for Mr Kushner include some of the President's outside advisers and some administration officials, the source said. Kelly, who is expected to leave around the end of this year, has been an important part of the president's daily structure.
Sanders said the White House continues to look at options and is monitoring the Senate's moves. "We ought not accept a CR until we exhaust every other opportunity to solve the appropriations". "There is some chance, but it would have to be a major development between the White House and the Democrats". "That's something that we would be able to support as a long as we can couple that with other funding resources". The feud over the US-Mexico border installation grows bigger and a government shutdown is not ruled out.
"I might say the administration is extremely flexible on this issue and we were obviously consulted before I sat down with Senator Schumer and Senator Leahy".
With Democrats near-unanimously opposing Trump's demand for $5 billion for the wall, Republicans lacked a path to pass it because they could not assemble the 60 Senate votes needed to advance a Homeland Security spending bill. The funding is all hung up over Trump's demands for $5 billion for the wall, which Democrats have rejected.
While Shelby would not entirely rule out a more global agreement between the House, Senate and White House, he was not optimistic. He was one of many Republicans who said that was the most likely option.
The two sides must come to some sort of budget deal by Friday or around a quarter of the government will shut down (CNBC reports that the affected agencies are "Homeland Security, [and] the departments of Transportation, Commerce, Interior, Agriculture, Housing and Urban Development and Justice").
The view from Democrats remains that their offers are on the table and it is up to McConnell and Trump to come back to them with another offer.