The Constitution was ratified more than two centuries ago, and in all that time no president had ever tested the limits of executive power enough to force the Supreme Court to rule whether he has lived up to the founders’ command that the laws be “faithfully executed.”

Until now.

When the justices convene Monday morning, they will hear what is shaping up to be the biggest case of the term, and perhaps one of the most consequential in a generation, as they consider whether President Obama has overstepped his constitutional powers by trying to grant a tentative deportation amnesty to up to 5 million illegal immigrants.

“In 225 years, the Supreme Court has never had occasion to ask the president whether he has reneged on his oath to take care that the laws are faithfully executed. However, with pens and phones replacing checks and balances, the Supreme Court is now poised to break new constitutional ground in order to preserve our embattled separation of powers,” said Josh Blackman, associate professor at the South Texas College of Law, who has followed the case from the start and filed amicus briefs opposing Mr. Obama’s claim of powers.

At issue is the Take Care Clause, which is what scholars call the Constitution’s charge to presidents to “take care that the laws be faithfully executed.”

That clause has been read to be both empowering to presidents, emboldening them with independent authority to see through the execution of laws, but also as a check — that, in the end, he carry out laws rather than write them.

For many, Mr. Obama’s deportation amnesty for illegal immigrants, announced in November 2014, strays too far into lawmaking.

At that time, Mr. Obama and Homeland Security Secretary Jeh Johnson announced a policy of granting nearly half of the 11 million illegal immigrants in the U.S. a proactive stay of deportation, along with the promise of work permits, entitling them to driver’s licenses, Social Security numbers and some taxpayer benefits.

Texas led 25 other states in suing, arguing that they suffered economic harm from the amnesty and that it would likely lead to even more illegal immigrants. The states said Mr. Obama broke administrative law and immigration law, and violated the Constitution.

Lower courts sided with the states and halted the amnesty on statutory grounds, and never reached the constitutional questions. But the justices, in what Mr. Blackman said was a first, asked both sides to also file briefs on the Take Care Clause.

Immigrant rights advocates say the case should be easy for the justices, pointing to smaller actions taken by Presidents Reagan and George H.W. Bush that they say paved the way for Mr. Obama. Those advocates say immigration legislation passed by Congress appear to approve of the “deferred action” power Mr. Obama used to halt deportations and grant work permits.

If Congress granted the president that kind of broad authority, then it’s legal and constitutional, said Brianne J. Gorod, chief counsel at the Constitutional Accountability Center.

“If the court concludes, as I think it will and should, that these executive actions are consistent with the laws passed by Congress, then it necessarily follows that they are consistent with the Constitution,” she said.

But John Yoo, a law professor at the University of California, Berkeley, said Mr. Obama’s intention in this instance wasn’t to carry out the law, but rather to undermine it by halting deportations — and granting a future get-out-of-jail card.

“The president gets to make the policy decisions about how the law is enforced. But he can’t take it all the way to say I’m not going to enforce it at all, and I think that’s where the line is crossed,” said Mr. Yoo, who served in the Bush administration and helped explore the limits of presidential war powers.

Mr. Obama has complicated his case by his own words. Before his 2014 announcement, he repeatedly and forcefully denied having the power to use deferred action so broadly. But after Congress refused to act on his immigration proposals, he reversed himself and claimed he did, after all, have the power.

Then just days after he announced the policy, he told an audience he “took an action to change the law” — a power that is supposed to be reserved for Congress.

Lower courts noted both of those statements in ruling against Mr. Obama, but Mr. Yoo said they are not likely to have much effect on the justices. He said it’s similar to how Chief Justice John G. Roberts Jr. didn’t give much credence during the Obamacare case to the president’s initial claim that the health care law wasn’t a tax.

Indeed, the justices eventually ruled that the reason Obamacare was constitutional is because it was a tax, no matter what Mr. Obama told voters.

Legal analysts said the case could decide the limits of presidential powers for years to come. Mr. Yoo said if the justices side with Mr. Obama, it opens the way for a future Republican president to use prosecutorial discretion to stop enforcing tax laws.

He also said Mr. Obama’s expansive interpretation of carrying out laws could frighten Congress away from dealing with the White House.

“He’s actually damaging the presidency for the future, because if you’re Congress in the future, and you’re working out some deal with the president, you can’t trust any deal you make with the president again,” he said.

A ruling is expected from the justices by the end of June.

But with the death of Justice Antonin Scalia, the eight-member court could deadlock 4-4. That would leave the appeals court’s decision in place, which would be a victory for Mr. Obama’s opponents.

A tie ruling, though, would also mean the court wouldn’t issue a definitive statement on Mr. Obama’s claims of power. That would leave for a future court the question of just how far a president can go to shape or, as Mr. Obama said, to “change” the law unilaterally.

Ms. Gorod said the justices may not even reach the big issues in the case. She said there is a good chance they throw out Texas’ challenge altogether, ruling that the state can’t show it’s has been harmed by the president’s action.

She said Chief Justice Roberts and Justice Anthony M. Kennedy were skeptical of similar claims in other cases.

Hispanic rights groups say the issue is already baked into the November elections no matter what the justices do and that voters are prepared to reward Democrats for backing Mr. Obama and punish Republicans for opposing him.

By a 5-1 margin, Hispanic voters say they are less likely to vote for a candidate who fought the amnesty. They also said they increasingly see Republicans as “hostile” toward Hispanics and view Democrats as “increasingly welcoming,” according to polling conducted by America’s Voice, a major advocacy group for immigrants’ rights.

Mr. Obama has argued that legalizing illegal immigrants and giving them a chance to work could pour billions of dollars into federal and state treasuries, bringing the immigrants into the above-ground economy, where their wages will be higher.

But the legal fight over the amnesty has proved damaging to the president, after the Homeland Security Department admitted it violated a lower court’s order halting the entire program early last year. Judge Andrew S. Hanen is still pondering whether to sanction the Justice Department for its behavior, saying the lawyers misrepresented some facts in the case.

On Friday, the Justice Department insisted it didn’t try to mislead the judge, saying there was an “unintentional miscommunication that occurred here.”

Judge Hanen issued an order in February 2015 keeping Mr. Obama’s broad amnesty from taking effect, and last year a federal appeals court, in a 2-1 decision, upheld that ruling.

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