The Supreme Court will hear arguments on Thursday in an extraordinary case that could alter the course of the presidential election by deciding whether former President Donald J. Trump’s conduct in trying to subvert the 2020 race made him ineligible to hold office again.

Not since Bush v. Gore, the 2000 decision that handed the presidency to George W. Bush, has the Supreme Court assumed such a direct role in the outcome of a presidential contest.

The sweep of the court’s ruling is likely to be broad. It will probably not only resolve whether Mr. Trump may appear on the Colorado primary ballot, but it will also most likely determine his eligibility to run in the general election and to hold office at all.

The case is just one of several involving or affecting Mr. Trump on the court’s docket or approaching it.

An appeals court ruled on Tuesday that Mr. Trump was not immune from prosecution for his role in the Jan. 6 attack on the Capitol, giving the former president until Monday to seek a stay of that ruling from the Supreme Court. And the justices have already agreed to decide on the scope of a central charge in the federal election-interference case against Mr. Trump, with a ruling by June.

Thursday’s case arose from a December ruling from the Colorado Supreme Court disqualifying Mr. Trump from the state’s Republican primary ballot based on Section 3 of the 14th Amendment. The provision was adopted after the Civil War to bar insurrectionists who had taken an oath to support the Constitution from holding office.

The provision says: “No person shall be a senator or representative in Congress, or elector of president and vice president, or hold any office, civil or military, under the United States, or under any state, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any state legislature, or as an executive or judicial officer of any state, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof.”

It adds, “But Congress may by a vote of two-thirds of each House, remove such disability.”

Mr. Trump has attacked the Colorado court’s ruling on at least a half-dozen grounds, though their unifying theme is that the election should be decided by the voters.

“The court should put a swift and decisive end to these ballot-disqualification efforts, which threaten to disenfranchise tens of millions of Americans and which promise to unleash chaos and bedlam if other state courts and state officials follow Colorado’s lead and exclude the likely Republican presidential nominee from their ballots,” Mr. Trump’s brief said.

The six Colorado voters who prevailed in the case in that state urged the justices not to give in to what they suggested were threats of political violence from a candidate they said had demonstrated a propensity for it.

“The thrust of Trump’s position is less legal than it is political,” the voters’ brief said. “He not-so-subtly threatens ‘bedlam’ if he is not on the ballot. But we already saw the ‘bedlam’ Trump unleashed when he was on the ballot and lost.”

Mr. Trump’s primary legal argument in the case, Trump v. Anderson, No. 23-719, is that Section 3 does not apply to him because the president is not among the officials covered by the provision. “The president is not an ‘officer of the United States’ as that term is used in the Constitution,” his brief said.

In addition, the brief argued, “Section 3 applies only to those who took an oath to ‘support’ the Constitution of the United States.” But, it continued, “the president swears a different oath set forth in Article II, in which he promises to ‘preserve, protect and defend the Constitution of the United States’ — and in which the word ‘support’ is nowhere to be found.”

Along similar lines, the brief also said that the presidency was not one of the offices from which oath-breaking officials were barred. “To accept the Colorado Supreme Court’s assertion that Section 3 includes the presidency,” the brief said, “one must conclude that the drafters decided to bury the most visible and prominent national office in a catchall term that includes low-ranking military officers, while choosing to explicitly mention presidential electors. This reading defies common sense.”

The brief also said that Section 3 disqualified people subject to it from holding office — not from seeking it. If the candidate were elected, the brief said, Congress could remove that disqualification before the candidate’s term began.

A Colorado trial judge ruled that Mr. Trump had engaged in insurrection but accepted his argument that Section 3 did not apply to the president or to the office of the presidency.

The Colorado Supreme Court affirmed the first part of the ruling — that Mr. Trump had engaged in an insurrection, including by setting out to overturn the result of the 2020 presidential election; trying to alter vote counts; encouraging bogus slates of competing electors; pressuring the vice president to violate the Constitution; and calling for the march on the Capitol.

But the majority reversed the part of the decision that said Section 3 did not apply to the presidency.

“President Trump asks us to hold,” the majority wrote in an unsigned opinion, “that Section 3 disqualifies every oath-breaking insurrectionist except the most powerful one and that it bars oath breakers from virtually every office, both state and federal, except the highest one in the land. Both results are inconsistent with the plain language and history of Section 3.”

The court added that states were authorized under the Constitution to assess the qualifications of presidential candidates. “Were we to adopt President Trump’s view,” the majority wrote, “Colorado could not exclude from the ballot even candidates who plainly do not satisfy the age, residency and citizenship requirements” of the Constitution.