But Lee and his allies are now urging the GOP to stop debate on that bill, acknowledging the Democrat-controlled Senate can easily gut the anti-Obamacare portion of the bill and send it back to the House.
After it became clear his Senate colleagues wouldn't back him, Cruz took to the Senate floor in a doomed filibuster – a symbolic gesture since the Senate is already slated to vote Wednesday.
"I intend to speak in support of defunding Obamacare until I am no longer able to stand," Cruz declared with Lee sitting to his left.
Lee spelled Cruz during the debate, allowing the Texas senator to sit while Lee pilloried the health care law and asked an elongated question.
The Cruz-Lee strategy on Obamacare was too convoluted for many Republican senators, who say they can't bring themselves to vote against a law they actually support. Hatch said the House bill slashes Obamacare funding, and that's why he'll vote for it.
"I support what the House did, and it's the only way to show support for it," Hatch said. "Let's put it this way: I'm not about to shut down the government and have Republicans take the blame for it. It's just that simple."
McConnell, who is up for re-election next year and faces a challenger on his right, says the House measure is the only path forward to stop a government shutdown and achieves the Republican goal of halting Obamacare.
"There's only one escape hatch that will fully help those trapped by this law, and that's full repeal," McConnell said on the Senate floor Tuesday. "And that's why I'm supporting the House-passed [resolution]. Not only does it defund this terrible law, it doesn't increase government spending by a penny, and it keeps the government from shutting down, which nobody wants."
McConnell, whose move provides cover for his GOP colleagues, says debating the resolution puts the focus where it belongs: on Democrats who will continue to support their signature, but controversial, health-care law.
Under the Senate's long-standing tradition, it takes 60 votes to bring up a bill for debate. While the GOP controls 46 seats, it will take only six Republicans to side with Democrats to get the bill past this hurdle. So far, well more than six Republicans have said they will vote for what is called "cloture" and more are expected to join this group throughout the day.
Once the Senate is debating the full measure, the Democrats, who control 54 seats, need only 51 votes to strip out language denying money for Obamacare and to send it back to the House, where it's unclear whether it will pass in time to stop a government shutdown before midnight Monday.
Cruz and Lee argue that they should try every method available to fight the law, even one that is confusing, such as using the cloture rules to stop debate on a bill they support. As Cruz told Fox News Sunday: "A vote for cloture is a vote for Obamacare."