But Cruz said it would be the fault of Democrats if the government shuts down.

"I do not believe we should shut down the federal government," Cruz said. "The only reason we might shut down the federal government is if President Obama and Majority Leader Reid decide they want to force a government shutdown."

Durbin then asked whether Cruz believes there are 60 votes in the Senate in favor of defunding ObamaCare. In response, Cruz seemed to admit that in the short term, defunding ObamaCare is not likely.

"I am not remotely Pollyannish, I am not remotely under the illusion that this is going to be a short-quick process that suddenly ObamaCare will be defunded," he said.

Cruz said defunding ObamaCare is a long-term process that starts with building support across the country. The next step is the House passage of the bill, but then unifying Senate Republicans, something he admitted has not happened yet.

"I believe every Republican should be unified, right now we're not," he said. "Right now there are divisions in the Republican caucus."

He said that once unified, Republicans may be able to eventually pull over a few Senate Democrats, particularly those in red states facing tough re-elections.

Durbin pressed Cruz on the idea that shutting down ObamaCare would hurt people like Judy, a woman in his home state who has never had insurance before but who can now access it through the exchanges. Cruz replied that the best way to cover people is to improve economic growth.

"I think your answer to Judy is you need a better job," Durbin shot back.

The two also sparred over what Cruz said was an Obama administration decision to exempt members of Congress and their staff from ObamaCare. But Durbin rejected that as incorrect.

"The senator is just plain wrong," he said. Durbin said the Obama waiver would allow the government to continue paying for 72 percent of the premiums for members of Congress and their staff. He said that is not a special exemption for Congress, but the continuation of a benefit that millions of Americans get through their own employers.