It would take at least 41 senators to block a spending bill, and Lee announced Thursday that he has the support of 11, including big name conservatives like Sens. Marco Rubio, John Thune and Rand Paul. He also has the backing of powerful outside groups, such as FreedomWorks, the Club for Growth and Heritage Action.
But so far Senate Republican Leader Mitch McConnell and House Speaker John Boehner have refused to endorse the plan. Sen. Orrin Hatch, R-Utah, isn't joining the effort and Sen. John McCain, R-Ariz., called it the kind of "shenanigans" most Americans are tired of.
No Republican senator has been more blunt than North Carolina's Richard Burr, who told Public Radio International that the shutdown threat is "the dumbest idea I've ever heard."
He said eliminating the Affordable Care Act is "not achievable" as long as Barack Obama is president and threatening to shut down the government over it puts Republicans in a bad political situation.
Chances slim to none? • Former Rep. Steve LaTourette, R-Ohio, a centrist Republican turned lobbyist, said Lee's gambit has almost no chance to succeed, though he gave him credit for "using every available tool that he has to try to make his point."
"If this leads to a conversation with the White House that's great, but if it leads to a government shutdown, I don't think that is a great place for the Republican Party to be," he said.
Burr and LaTourette remember the last government shutdown in 1996, when President Bill Clinton came out in far better shape politically than the Republicans did.
Lee and his supporters are aware of this criticism and they believe they can pin the blame for any potential shutdown on the Democrats for standing next to a law that has more detractors than supporters.
Rubio wrote a column for Fox News saying that it's Obama who deserves the blame for a law that may increase premiums on people who buy insurance on the open market and cause small businesses to shift some workers from full to part time.
"It is time to admit that Obamacare isn't going to work, decide not to waste a single cent more on it, and replace it with market-based reforms that will give people more health insurance choices and options," he wrote. "It's still not too late. This September may be our last best chance to do anything about this disastrous law."
And Lee appeared on Laura Ingraham's radio show where he said: "I don't think Harry Reid has the political wherewithal right now, given the dwindling support for this legislation by the American people, to shut down the government for Obamacare. He doesn't want to go down in flames defending this law in that way."
Reid, the Senate Majority Leader, had his own response to Lee, predicting that Utah's junior senator couldn't get enough support to even try a shutdown.
'Dreamland' • "He's living in a dreamland," Reid said in an interview with PBS Newshour. "Even Republicans won't agree with what he's trying to do."
Reid called Lee's effort "senseless and illogical," noting he wants to get rid of the health-reform law, but keep some its most popular provisions such as banning insurance companies from denying coverage for pre-existing conditions and prohibiting lifetime limits on benefits.
Some Democrats believe Lee is doing them a favor by prodding Republicans to join his effort.
"As a Democratic strategist, I want to thank Senator Lee for proposing such an extreme measure," said Jim Manley, a former Reid spokesman. "Stuff like this really helps us to paint Republicans as out of touch and captive of the far right wing. What he is suggesting is a recipe for disaster. It will rattle the market, hurt the economy and possibly do harm to peoples' 401(k)s."
Norm Ornstein, a congressional scholar employed by the conservative American Enterprise Institute, said it wasn't surprising that Lee would threaten a shutdown, calling him a "take-no-prisoners extremist senator from Utah." But in a column in the National Journal, Ornstein said it is "depressing" to see that the Senate's third highest-ranking Republican, Sen. John Thune of South Dakota, signed on.
Ornstein said the effort to undermine the law, passed by Congress and upheld by the Supreme Court, is unprecedented and "sharply beneath any reasonable standards of elected officials with the fiduciary responsibility of governing."