"I believe we should stand our ground," said Cruz, who already was trying to blame Obama and his Democratic allies if the government shuts down.
Sen. Claire McCaskill, a Missouri Democrat, said Cruz's efforts were destructive and self-serving as Cruz eyes a White House campaign.
"I cannot believe that they are going to throw a tantrum and throw the American people and our economic recovery under the bus," she said.
"This is about running for president with Ted Cruz. This isn't about meaningful statesmanship," she added later.
The wrangling over the budget comes as lawmakers consider separate legislation that would let the United States avoid a first-ever default on its debt obligations. House Republicans are planning legislation that would attach a 1-year delay in the health care law in exchange for ability to increase the nation's credit limit of $16.7 trillion.
Obama, speaking to political allies on Saturday evening, showed little patience for the GOP efforts to undermine his legislative accomplishment by either avenue.
"We will not negotiate over whether or not America should keep its word and meet its obligations," Obama told the Congressional Black Caucus Foundation dinner. "We're not going to allow anyone to inflict economic pain on millions of our own people just to make an ideological point."
Congress doesn't seem eager to help Obama, although there are deep divides — both between parties and within them — over who deserves blame.
Rep. Tom Graves, R-Ga., said the goal was to defund the president's health care legislation for at least one more year if not forever.
"We do have eight days to reach a resolution on this, and I propose an idea that kept the government operating and opened for an entire year while delaying and defunding Obamacare for a year so that we could work out those differences," Graves said.