The Senate on Monday failed for a fifth time to pass spending measures reopening the government.
Competing proposals by Democrats and Republicans were both shot down, failing to meet the 60-vote threshold required.
President Donald Trump said earlier in the day that another failed vote would trigger mass layoffs. Thousands of federal employees were furloughed or ordered to work without pay when funding for their agencies ended five days ago.
But he hinted that he was open to trying to end the impasse and possibly strike a deal with Democrats who are sticking to demands that the legislation address healthcare. Republicans are pressing for a clean funding bill.
The Democratic-led bill to extend the government's funding failed first on Monday with a 45-50 vote. Its Republican counterpart then fell short at 52-42.
Soon after the vote, Trump took to social media to blame Democrats for shutting down the government.
Democrats have held out on supporting the Republican-led proposal because they say it undercuts medical access for lower-income Americans.
In separate interviews with NBC's Meet the Press on Sunday, the House's top Democrat, Hakeem Jeffries, and top Republican, Speaker Mike Johnson, each blamed the other's party for the continued stand-off.
Leavitt on Monday called on Democratic lawmakers to relent. She stated, There's nothing to negotiate. Just reopen the government.