The Supreme Court has flirted with a flurry of Trump administration matters in recent months, but the battle over the president’s sweeping global tariffs will put the justices directly on the spot over a centerpiece of his economic agenda.

President Trump has been on an all-out blitz to expand executive power, including by declaring a series of emergencies on top of policy priorities that he says allow him to bypass normal procedures and take unilateral action. Citing national security, he upended global markets by declaring a trade emergency to impose 10% baseline tariffs on virtually all countries—and more on some nations. He also claimed an opioid trafficking emergency to direct additional tariffs to Canada, China and Mexico.

Across three different courts, 15 judges have weighed Trump’s tariff maneuvers—and 11 of them, appointed by presidents of both parties, have found he acted without legal support. The most consequential of those decisions came late Friday from the U.S. Court of Appeals for the Federal Circuit, which rejected the tariffs and gave the president a mid-October deadline to appeal to the Supreme Court before the ruling takes effect.

The White House always expected the dispute to be settled at the high court, and it is betting that the court’s conservative majority, one that stands to the right of many lower courts that have ruled against the administration this year, will affirm Trump’s sweeping assertion of his own authority.

There are reasons its optimism makes sense.

To date, the administration has sought preliminary relief from the Supreme Court in more than 20 cases in which lower court judges temporarily blocked the White House’s plans. In many of them, the high court gave the administration what it asked for, in orders that provided little—if any—explanation.

Often over the dissent of its three liberal members, the court has granted Trump’s emergency requests to fire federal officials , deport some classes of immigrants with minimal due process , withhold congressionally appropriated research and education funding and expel transgender service members from the armed forces, among other matters.

The court’s orders in those cases may be provisional, but they let the administration have its way for months or perhaps years while litigation proceeds.

The tariff case, however, may not be so easy to predict.

Not only does Trump’s policy break with longstanding Republican Party orthodoxy in favor of free trade, his choice to implement it unilaterally rather than through legislation has created legal vulnerabilities that plaintiffs have been quick to exploit.

The parties challenging the tariffs go beyond the progressive groups and Democratic-led states involved in most suits against the administration. Here, small businesses , trade associations and right-leaning advocacy groups have all filed briefs complaining that the tariffs are illegal, and their arguments rely on legal doctrines the Supreme Court previously has cited to strike down excesses it found in Biden administration policies.

A final decision still could be many months away. Because the administration still can collect tariffs for now, the White House faces no real urgency to move faster than the mid-October deadline to file an appeal with the Supreme Court. It could be December or January before the court agrees to take up the case, which would likely put oral arguments in the winter or early spring. A decision likely would be weeks or months after that.

Central to the case could be the “major questions doctrine,” a term the Supreme Court coined for expansive presidential policies the justices feel go beyond what lawmakers had in mind when adopting broadly written statutes. So far, it has wielded the doctrine to shoot down liberal policies.

The rule applies when reviewing policies that entail “vast economic and political significance,” Chief Justice John Roberts wrote in a 2022 decision voiding the Clean Power Plan , an Obama-era initiative to reduce greenhouse gases that could have forced some coal and gas plants to shut down in favor of those fueled by renewable sources or natural gas.

The court’s conservative majority used similar reasoning to throw out a moratorium on evictions, workplace measures to counter Covid-19 and across-the-board forgiveness of student-loan debt the Biden administration had sought to implement under presidential emergency powers.

The tariff case could force the court to confront competing strains of its jurisprudence. While the court has sought to draw firm lines that prevent a president from adopting sweeping policies that intrude on the lawmaking function of Congress, it also has sought to remove impediments that interfere with his ability to exercise his constitutional powers to the fullest.

Mark Graber , a law professor at the University of Maryland, said there are several reasons that the Supreme Court is likely to take a critical look at Trump’s position in the tariff case. For one, he said, there is no indication that the statute Trump invoked contemplated a president constantly imposing and adjusting tariffs in response to chronic issues like drug smuggling and trade imbalances.

“When Trump has claimed powers that are purely executive, the court has basically said, ‘You’re right,’” Graber said. “Where the court has sometimes pushed back is when Congress has been quite clear and Trump is claiming a power to ignore Congress.”

The 1977 law Trump cited to impose the tariffs, the International Emergency Economic Powers Act, doesn’t mention tariffs, and until Trump no president had invoked the statute to justify import taxes.

On the other hand, there is little legal precedent on the executive’s unilateral power to impose tariffs, since the last president to leverage them in an attempt to remake the global trading environment was Richard Nixon , in 1971. That leaves ample room for the court to consider Trump’s arguments.

Perhaps the president’s best hope is a dissent penned in Friday’s ruling—by a judge appointed by President Barack Obama . Judge Richard Taranto , joined by three other judges, said Ieepa uses broad language to provide flexibility in the tools available to the president to address unusual and extraordinary threats.

“We know of no persuasive basis for thinking that Congress wanted to deny the President use of the tariffing tool, a common regulatory tool, to address the threats covered by IEEPA,” Taranto wrote.

Appearing Sunday on Fox News, Peter Navarro , White House senior adviser on trade and manufacturing, said the dissent “presents a very clear road map to how the Supreme Court can rule in our favor.”

Write to Jess Bravin at Jess.Bravin@wsj.com