The discovery of classified material at former Vice President Mike Pence’s Indiana home tests the standard Attorney General Merrick Garland laid out when appointing two special counsels to examine similar breaches by President Biden and former President Donald Trump, citing their likely 2024 presidential runs.

Mr. Pence is also considering a White House bid. While Mr. Garland hasn’t said whether he will have a special counsel review documents found at the former vice president’s home, the prospect illuminates the dilemma he now faces in referring politically unpredictable probes to outside prosecutors over whose decisions he has little control.  

Mr. Garland “is between a rock and a hard place now,” said Stephen Saltzburg, a former Justice Department official and law professor at George Washington University. “If he doesn’t appoint a special counsel the Democrats will say ‘why not?’ If he does, people will say ‘we’ve never had so many special counsels!’ If I were the attorney general, I would be rueing the day I appointed the first special counsel.”

The FBI and Justice Department’s National Security Division are reviewing how the documents ended up in Mr. Pence’s home, people familiar with the matter said, adding to the growing docket of politically fraught cases involving classified information confronting Mr. Garland.

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Mr. Garland could take action short of appointing another special counsel, former prosecutors said, such as adding the review of Pence documents to the remit of Robert Hur, whom he named earlier this month to head the inquiry into documents found at Mr. Biden’s home and former office. 

A Justice Department spokeswoman declined to comment on Wednesday.

Mr. Garland, a former federal judge whom Mr. Biden selected in part for his pledge to keep the Justice Department free of partisan influence, at first resisted naming a special counsel to investigate the handling of classified documents found at Mr. Trump’s Mar-a-Lago resort, people close to him have said, because he firmly believed the department was capable of fairly handling the probe on its own. 

But when Mr. Trump announced his White House bid in November, the attorney general said the political environment left him no choice but to appoint an outsider, former war-crimes prosecutor Jack Smith, to lead the inquiry.

After Mr. Biden’s legal team began finding classified material at his former office and Delaware home around the same time, some legal experts said Mr. Garland’s adherence to department norms that require like cases to be treated alike left him little choice but to appoint Mr. Hur to examine the matter. 

Such norms “are essential for us to continue to assure that we adhere to the rule of law,” Mr. Garland said this week when asked about the parallel special counsel probes. “These mean, among other things, that we do not have different rules for Democrats or Republicans.”

Mr. Garland also said the prospect of making decisions in an investigation involving his own boss qualified as an “extraordinary circumstance” necessitating a special counsel. 

Republicans accuse Mr. Garland of a double standard in his treatment of Messrs. Trump and Biden, citing the FBI’s warrant-backed search of Mar-a-Lago in August. “The department’s actions here appear to depart from how it acted in similar circumstances,” the GOP co-chairs of the House Judiciary Committee said shortly after Mr. Hur’s appointment.   

White House officials have said they are fully cooperating with the Justice Department’s request to not disclose anything about the process in public unless told otherwise. Such constraints have resulted in the White House drawing criticism for offering incomplete information about the searches and a fragmented timeline of events.

Asked if a special counsel should be appointed to investigate the classified documents found at Mr. Pence’s home, the White House said it was up to the Justice Department and reiterated Mr. Biden’s pledge to respect the department’s independence.

“We will not politically interfere,” White House press secretary Karine Jean-Pierre told reporters Tuesday. 

Differences among the cases mean they should be treated differently, said Mr. Saltzburg, who served as associate independent counsel in the Iran-Contra probe in the 1980s, noting that most cases involving classified information don’t result in criminal charges, let alone the appointment of a special counsel.

Mr. Pence’s lawyers have said he took the documents by mistake and was unaware he had them.

In the Mar-a-Lago matter, Mr. Trump’s team for months ignored demands from federal authorities that documents be turned over, and the probe escalated over more than a year to include a criminal investigation into possible obstruction, among other potential crimes. 

Mr. Biden and his advisers say they are cooperating with the Justice Department and FBI, inviting agents to search his Wilmington home last week. That search was at least the fifth that has turned up classified material in places used by Mr. Biden.

While the special-counsel appointments reflected Mr. Garland’s oft-stated goal of insulating the Justice Department from charges of political bias, some legal experts said tapping additional outsiders may have the reverse effect of demonstrating that politics are unavoidable for an agency buffeted by partisan pressures from both sides.

“I think Garland is doing a good job of trying to insulate the Department of Justice from political corruption and headwinds,” said Rory Little, a professor at UC College of the Law, San Francisco, who worked with Mr. Garland in the Justice Department in the 1990s. “I don’t think he anticipated the political fallout he now faces.”

The dynamic of having at least two parallel investigations, however, may also make it more challenging for Mr. Garland to demonstrate to the public that the probes are being handled equitably.

“These two are going to sit on their own merits and take place in their own timing and come to entirely separate conclusions without regard for one another,” said former Deputy Attorney General Jamie Gorelick, to whom Messrs. Garland and Little reported when they served under President Bill Clinton.

The side-by-side special counsels also come as outsiders are handling other politically sensitive probes.

John Durham, a former U.S. attorney in Connecticut, is continuing as special counsel to investigate the FBI’s handling of a 2016 inquiry into Russia’s interference in that year’s presidential election and any links to the Trump campaign. And David Weiss, the Trump-appointed U.S. attorney in Delaware, is examining whether to bring a criminal case against Mr. Biden’s son, Hunter Biden, related to his tax dealings and a gun purchase. Mr. Weiss isn’t a special counsel but while other Trump U.S. attorneys left at the start of the administration, Mr. Garland kept Mr. Weiss on to similarly avoid the appearance of a conflict.

—Sabrina Siddiqui and C. Ryan Barber contributed to this article.

Write to Sadie Gurman at sadie.gurman@wsj.com