The Senate is currently adjourned, raising fears that the situation could drag on and threaten hundreds of thousands of jobs, as well as risk costing the US economy billions in lost output.
At a White House briefing on Wednesday afternoon, Vice-President JD Vance made a rare appearance alongside Press Secretary Karoline Leavitt and accused opposition Democrats of playing political games.
"If they are so worried about the effect this is having on the American people, and they should be, what they should do is reopen the government, not complain about how we respond," he said.
Leavitt, meanwhile, indicated that mass job cuts could happen before the end of the week. Pressed by reporters about timeframes, she said "two days, imminent, very soon" and promised further updates.
"Sometimes you have to do things that you don't want to do," Leavitt explained, adding that "Democrats put us in this position".
Trump previously warned that a prolonged shutdown could present him the opportunity to make irreversible cuts to "Democrat things".
Writing on his Truth Social site on Wednesday evening, he added: "Republicans must use this opportunity of Democrat forced closure to clear out dead wood, waste, and fraud. Billions of Dollars can be saved."
His administration announced the same day that $18bn (£14bn) would be withheld for infrastructure projects in New York City - the home city of the Democratic leaders in the House and Senate.
Vance's comments at the White House represented the latest dig in what has been a bitter blame game between both parties, with the top Senate Democrat Chuck Schumer earlier accusing Republicans of trying to "bully" Democrats into accepting their funding plan.
Democrats want to secure guarantees on healthcare funding before they agree to a spending deal, while Republicans want to use a temporary stop-gap measure to keep the government open until mid-November and funded at current levels.
Schumer and his allies say they allowed the government to shut down in an attempt to negotiate to save healthcare benefits for lower-income Americans. They have said efforts to negotiate with Republicans over these benefits have so far been unsuccessful.
“Why are they boycotting negotiations? I've never witnessed this in my life," Senator Chris Murphy, a Democrat from Connecticut, said of Republicans. “The matter is the government will open when Republicans get serious about it talking to Democrats.”