At that time, most people will be required to have insurance or pay a fine, a law known as the individual mandate. And for the minority of people who don't get coverage through their jobs, they'll have new online insurance exchanges to shop in, with those on the lower end of the income scale getting federal subsidies.
President Barack Obama has touted his signature law as a major step forward, one that he says will help lower costs over the long run and allow people to keep insurance when they lose jobs.
But Republicans argue the law is too costly, too unwieldy and will result in immediate premium increases. They also staunchly object to the individual mandate, which the Supreme Court upheld as a tax.
"If Republicans in both houses simply refuse to vote for any continuing resolution that contains further funding for further enforcement of Obamacare, we can stop it," Lee said on Fox News. "We can stop the individual mandate from going into effect."
Like almost all congressional Republicans, Lee has advocated for a full repeal of the law known formally as the Affordable Care Act. But after the Supreme Court decision and Obama's re-election, Lee conceded that a repeal was unlikely.
He's shifted tactics, instead focusing on the funding of the implementation. He says his attempt would have no bearing on the parts of the law that have already been put in place, which include such popular items as free preventive care, allowing people up to age 26 to remain on their parents' health plan and a ban on insurance companies denying coverage for people with pre-existing conditions.
"We are talking about further enforcement and implementation of Obamacare," Lee said on Fox News. "We are talking about those [provisions] that haven't kicked in yet. We are talking about the individual mandate, the exchanges, the exchange subsidies and so forth."
Lee launched the effort last week in a speech on the Senate floor. He criticized the president for unilaterally delaying a requirement that larger employers offer insurance or pay a fine and another requirement that people prove their income before receiving federal help to pay their health-care premiums.
"In other words, if the president won't follow it, the American people shouldn't fund it," he said in that speech.
Since then, he has granted interviews to conservative radio and television hosts to pitch the plan. Lee declined an interview request from The Salt Lake Tribune.
A spokesman for the senator said cutting off funding would also cut off enforcement for the existing parts of the law, including the pre-existing condition portion. Lee would then push for a bill to make insurance portable, which would prevent people from losing coverage when they change jobs. That's a plan Republicans have long championed, though most Democrats consider it inadequate.
The showdown could result in the first government closure since 1996, though Lee's office considers that unlikely and, if it happens, the senator would attempt to blame Democrats for failing to abandon Obamacare.
But before that would happen, Lee would need to gain the support of more than 40 senators to block a continuing resolution in the Senate. Even if he does have more than a dozen supporters, he has a way to go to achieve the stated goal.
Sen. Orrin Hatch, R-Utah, is not among those supporters, though he has led numerous attempts to dismantle the health-care law. A Hatch spokesman said he is reviewing "every way to repeal it."
Lee expressed confidence Monday the effort will be successful.
"The number is growing. I expect to get more people signing on this week," he said on Fox News. "This movement is going to build."