Washington (CNN) As lawmakers return to Washington after Thanksgiving, Congress is careening toward a spending fight over President Donald Trump's campaign promise of a border wall that threatens to trigger a partial government shutdown.

The clock is ticking with funding for several government agencies, including the Department of Homeland Security, set to expire on December 7.

Trump has asked for $5 billion in wall funding and is ratcheting up the pressure for Congress to allocate money for the wall, which he promised on the campaign trail would be paid for by Mexico. But Democrats are pushing back against that request and it exceeds the $1.6 billion in an existing Senate spending bill that would pay for fencing along the southwestern border.

Making efforts to reach a deal more complicated, Democrats are eying a potential demand of their own in an effort to pass a measure to protect special counsel Robert Mueller's investigation into Russian interference, though they are so far signaling they won't shut down the government over it.

Senate Democratic Leader Chuck Schumer on Monday called for the Mueller bill to be attached to spending legislation if Senate Majority Leader Mitch McConnell won't bring it to a floor vote, but stopped short of issuing an ultimatum or warning that there could be a shutdown over the issue.

"Nobody has said it that way," Sen. Patty Murray, a top appropriator and senior member of Senate Democratic leadership, told CNN when asked if Democrats would draw a red line over the spending bill if the Mueller plan is not included in it. "I think everybody feels very strongly we need to get the bill done -- how it gets done hasn't been the issue."

Asked if a shutdown could occur over the special counsel bill, Murray said: "There are so many others in the way -- I don't see that being one."

Top congressional Republicans meanwhile are expressing confidence they can reach a spending deal that would avert a shutdown, but are nevertheless acknowledging that a partial shutdown could occur even as they say they don't want it to happen.

"Where there's a will, there's a way," Sen. John Cornyn, the No. 2 Senate Republican, told reporters when asked if he believes Congress can pass the remaining spending bills ahead of the upcoming deadline. But, he added, "There are a lot of variables to deal with ... I don't think the Democrats want to shut down the government over the border security issue. But they might. We'll see."

Republican Sen. John Kennedy of Louisiana said that "a government shutdown would be stupid," and that it would be "a grave mistake" if Democrats push to add the Mueller bill to must-pass spending legislation.

"That very well could lead to a government shutdown," he said.

Trump keeps the pressure on Congress

So far, Trump is ratcheting up the pressure. The President started the week by calling on Congress to fund his signature campaign promise and said earlier this month it could be a "good time" for a shutdown if he doesn't receive the funding he wants.

Senate Appropriations Chairman Richard Shelby told reporters on Monday that the President has said that his $5 billion request is a red line.

Asked if Trump would back off the $5 billion request, the Alabama Republican told CNN: "Not from my conversation with him." But the senator said that there could be a way to get close to the $5 billion request "depending on how the money is configured" since some money already appropriated by Congress has yet to be spent on the wall.

"It depends on how it's configured," Shelby said. "There's some money owed already and we talked about $1.6 billion, the House is at $5 billion -- it depends on how it's configured though."

Republicans managed to avoid a showdown over the border wall before the midterm elections by passing a spending package that funded much of the government, but effectively punted the border wall fight until after the election was over.

Republicans will only control both the House and the Senate for a little while longer, until the new Democrat-led House of Representatives takes over in January. But even under Republican control, spending bills must clear a 60-vote hurdle to pass in the Senate and Republicans currently have just 51 seats, meaning any funding legislation would still need some Democratic support to pass.

Democrats may be emboldened by the party's midterm victory to fight even harder against the President's demand for wall funding.

"The President is going to have to make a decision whether he wants to shut down the government after just having his hat handed to him in the election," Democratic Sen. Chris Murphy of Connecticut told reporters on Monday, predicting that it will be "hard to convince" Democrats to "give in" to what he described as "ridiculous demands to build a border wall that nobody wants."

But sat the same time, some influential Democrats are signaling there's still a chance a deal is cut to keep the government open by December 7 that would resolve all the outstanding issues, including Trump's border wall.

Asked if he could reach a deal by the deadline, Sen. Patrick Leahy, the top Democrat on the Appropriations Committee, said: "I could."