The developments add a new and unexpected layer to the complex relationship between the U.S. and Iran, which had been radically shifted by President Donald Trump’s complaints about the nuclear accord and Iran’s growing influence in the Middle East as that of Islamic State declines.

President Donald Trump tweeted “TIME FOR CHANGE!” on Monday in strong support of the Iranian demonstrators. His administration’s outspoken stance has ignited a broad debate over how forcefully the U.S. should respond to the protests inside Iran and whether expressions of American support for the demonstrators would help or hurt their cause.

“We are encouraging all nations around the world to publicly condemn the government violence and to support the legitimate, basic rights of those protesting,” said Brian Hook, the State Department’s director of policy planning, said in an interview.

“We know that the IRGC plays a big role in the decisions and actions of the government,” he added, using the acronym for the Revolutionary Guards.

The protests are the largest in Iran since demonstrations during the country’s Green Revolution in 2009, with much of Iranians’ ire focused on economic concerns. As protests build, the discontent has widened to target Iran’s ruling system.

The Obama administration was criticized for being slow to show support for those leading 2009 demonstrations. Veterans of that administration say Mr. Trump’s vocal support for the opposition could backfire by enabling the Iranian regime to portray demonstrators as American agents.

“Instead of this being been seen in Iran as an issue of the regime ruining the economy and not serving the people well, it could be framed by the Iranian government as the United States seeking to decide how Iranians should govern their country,” said Philip Gordon, who served on President Barack Obama’s National Security Council. “That framing is not helpful. “

Mr. Hook pushed back on the notion that energetically backing Iranians’ right to publicly express their views would give ammunition to the regime.

“It doesn’t matter what we do, they will blame us,” he said. “For us, this is not a complicated question. We want to take a position with moral clarity and let the protesters know they’re not alone.”

In an effort to mobilize international support for its stance, Trump administration officials have been working on a joint statement with the U.K., France, Germany and Italy that would urge Iranian authorities to respect their citizens’ rights and not to use violence against the protesters. The Trump administration is also eyeing action at the United Nations in the coming days, officials said.

“We are actively collecting all information on human-rights abuses in Iran against peaceful protesters,” Mr. Hook said.

The question of how to respond to protests in Iran has long been a policy challenge for the U.S. In 2009, the Obama administration was caught off guard when millions of Iranians protested the allegedly fraudulent re-election of President Mahmoud Ahmadinejad.

Some U.S. officials pressed Mr. Obama to publicly back the fledgling Green Movement, arguing in Oval Office meetings that it marked the most important democratic opening since the 1979 Islamic revolution. But Mr. Obama, who was involved in developing a secret diplomatic outreach to Supreme Leader Ayatollah Ali Khamenei, was cautious and slow to respond.

Officials who served at the time said Mr. Obama wanted to maintain an opening for future talks with Iran and was also influenced by the mixed messages sent by Green Movement leaders. Some of those leaders urged Mr. Obama to publicly warn Mr. Khamenei against a crackdown, while others said too active an American role would give the ayatollah and his backers an excuse to paint the opposition as American lackeys.

Some experts said the Trump administration was right to speak out about the current protests but said its statements needed to be more disciplined and part of a coordinated strategy.

“What’s needed are carefully crafted statements of solidarity with the Iranian people rather than freewheeling tweets from Trump encouraging protesters, which would be counterproductive,” said Karim Sadjadpour, an expert on Iran at the Carnegie Endowment for International Peace.

“What the U.S. government does, rather than says, is much more important,” Mr. Sadjadpour added. “We should make clear to companies and countries around the world that those complicit in providing Tehran the means and technology to inflict violence and a communications blackout on Iranian society will face censure from the U.S.”

Mark Dubowitz, the chief executive officer of the Foundation for Defense of Democracies, a think tank that has been sharply critical of the Iranian government, said the Iranian regime will try to blame the U.S. for the protests regardless of what action Washington takes.

“Western governments should make it clear that the regime will be held responsible and will pay a price for any bloodshed,” Mr. Dubowitz said.

Mr. Trump has been outspoken. “The great Iranian people have been repressed for many years,” he said in a tweet Sunday. “They are hungry for food & for freedom. Along with human rights, the wealth of Iran is being looted.”

Mr. Trump faces a deadline late next week to tell Congress whether the nuclear deal with Iran is in America’s interest. Last year, he said it wasn’t. The president is expected to again withhold support for the accord while extending sanctions relief guaranteed under the deal and allowing the agreement to remain in place. But the Iranian government’s actions against the protesters could color Mr. Trump’s deliberations, some experts say.

Some U.S. allies have issued their own statements on the protests in Iran.

Boris Johnson, the British foreign secretary, said Sunday on Twitter that he was “watching events in Iran with concern” and that “citizens should have the right to demonstrate peacefully.”

German Foreign Minister Sigmar Gabriel on Monday called on Iran’s government “to respect demonstrators’ freedom of assembly and their right to give voice to their opinion freely and peacefully.” He said the confrontations of the past days made it “all the more important for all sides to refrain from taking any violent action.”

Catherine Ray, a spokeswoman for European Union foreign policy chief Federica Mogherini, said the EU expected the right to peaceful demonstration and freedom of expression to be guaranteed, in keeping with statements from Iran’s president.