But Republicans, such as Sen. David Perdue of Georgia, don't want Trump to throw away what they see as his strongest bargaining chip.
Perdue balked when asked Wednesday on CNBC's "Squawk Box" about the calls from Democratic leaders, including House Speaker Nancy Pelosi and Senate Minority Leader Chuck Schumer, to reopen the government and continue separate border security talks.
"Right now, what the Democrats are saying is, 'Well look, give us what we want and maybe we'll come back in two or three months and talk about what you want.' That's not the way negotiations work in the real world," Perdue said.
When pressed on the Democrats' offer to reopen the government, Perdue suggested that such an agreement would undermine Trump's efforts to pressure his opponents into funding a border wall.
"I understand that," Perdue said. "But then, where's the leverage for the president in terms of getting them to move on the issue of the day?"
Perdue's reference to the shutdown — now in its 26th day with negotiations at a stalemate — as "leverage" for Trump comes as federal workers are feeling its effects more acutely than ever.
With about 800,000 government workers furloughed or continuing to work without receiving regular paychecks, the impact of the shutdown is spilling over into everyday life for other Americans, as well. At airports, for instance, passengers are reportedly facing huge delays and limited services as Transportation Service Administration workers pass their first pay period without receiving a check.
New forecasts from the Trump administration show that the partial shutdown could cost the economy twice as much as originally thought. An official told CNBC on Tuesday that the administration doubled its original estimate to a 0.1 percentage point subtraction from growth every week.
Perdue spokeswoman Casey Black said in a phone call after this story was originally published that the senator is in favor of ending the shutdown and is trying to get to a solution on border security.
Trump told Schumer in December that he would be "proud" to shut down the government in pursuit of constructing a wall along the U.S.-Mexico border — the central promise of his 2016 presidential campaign. But he quickly reverted to blaming Democrats for refusing to allocate money toward a wall as part of any spending deal to fund the whole government.
Republicans have been quick to point out that Democrats have in the recent past been willing to provide much more money for a border wall as part of broader immigration reform efforts, none of which came to pass. Trump is currently asking for $5.7 billion, which the Office of Management and Budget recently estimated would help build fewer than 250 miles of wall on the nearly 2,000-mile border.
Schumer and other Democrats, however, argue that holding firm against Trump's wall proposal in this standoff will prevent the president from using government shutdowns as a negotiating tactic in the future.
"The last thing you want to do is reward the president's behavior of holding all these folks hostage," Sen. Mark Warner, D-Va., vice chairman of the Senate Intelligence Committee, said Wednesday on "Squawk Box."
"If this tactic at all proves to be successful, you'll see it again on debt ceiling, see it again on the end-of-the-year spending bills," Warner added. He noted "the amount of economic hurt that is taking place at the family level and on the general economy level" in Virginia and Washington, D.C., already.