Swelling numbers of protesters on the streets of Cairo dismissed a resignation offer from Egypt’s cabinet on Monday and said they would be satisfied only if the country’s military leaders agree to cede power by next summer.

With another major march scheduled for Tuesday, it was clear that the violent clashes that began over the weekend are posing the most serious challenge to the military’s hold on power since it took control of the country in February.

The three days of unrest have left at least 33 protesters dead. The ruling Supreme Council of the Armed Forces issued a statement saying that it was watching the “escalating events” with “extreme caution and sorrow.”

It remained unclear whether the protests would disrupt parliamentary elections scheduled for next week. Victoria Nuland, a spokeswoman for the State Department, said that the United States was “deeply concerned” about the bloodshed and urged all parties to focus “on holding free, fair and peaceful elections as scheduled” on Monday.

As the resignation offer from the military-backed cabinet was announced Monday night, Egyptians thronged into Tahrir Square chanting slogans against the military council. Riot policemen for a third straight day battled rock-throwing protesters with tear gas and rubber-coated bullets.

The cabinet, led by Prime Minister Essam Sharaf, called the recent violence “unfortunate” and said it felt “politically responsible.” But it was unclear whether the cabinet’s offer to step down was intended to appease activists or register opposition to the rough treatment of protesters, which was carried out by security forces with military backing.

There were conflicting media reports about whether the generals had accepted the resignation, and a military spokesman reached by phone said he could not say whether the generals had accepted it.

But activists called the cabinet’s resignation offer insignificant.

“We don’t care if the government resigned; we want the head of the snake, not the body,” said Abdalla Waleed, 20, a blogger who has been protesting in Cairo’s central Tahrir Square.

The country’s military generals were hailed as heroes when they sided with demonstrators and pushed President Hosni Mubarak out of power in what amounted to a soft coup. Over the past nine months, however, the generals have tried thousands of civilians in military courts and expanded the use of a despised emergency law that gives the government sweeping powers to detain people, worrying some Egyptians that they had traded one autocratic regime for another.

Political leaders across the ideological spectrum appeared to coalesce Monday around a list of demands. Key among them was that military leaders cede power to elected officials before the summer, rather than sticking to the current transition time frame, which could leave them in control for up to two more years.

The Revolutionary Youth Coalition, a group of pro-democracy organizations, called for a “million-man march” Tuesday. Representatives of the group have asked the generals in charge to hand power to a national unity government led by Mohamed ElBaradei, the former head of the International Atomic Energy Agency and a Nobel Peace Prize winner. ElBaradei has been in talks with the military leadership, said Shady Ghazali Harb, a leading member of the coalition.

In an apparent nod to protesters, the military announced a new “treachery” law Monday to stop former members of Mubarak’s National Democratic Party from running in the parliamentary elections. But the law, which activists said was too little, too late, would apply only to those convicted of political corruption.

The military council also said that it had ordered an investigation of the causes of the clashes and would take legal action “against anyone who is proven to have been involved in the events.”

The Muslim Brotherhood, Egypt’s most organized political force, said Monday that its Freedom and Justice Party had suspended campaign activities because of the unrest. But the party insisted that elections, the country’s first since Mubarak’s ouster, be held as scheduled. Several other candidates suspended their campaigns over the weekend.

The Brotherhood’s party also said in a statement late Monday that it would not participate in further protests, worried that the swelling revolt could undercut the elections, in which it expects to do well.

The pressure against the ruling council began to build Friday, when the Muslim Brotherhood and Salafists, who follow a rigid form of Islam, took to the streets in large numbers to demand a quick transition to civilian rule and limits on the military’s power.

Most Islamists had left Tahrir Square by Friday night, but a small group of protesters remained, and their ranks have continued to swell. They have been battling riot police since Saturday, when authorities attempted to prevent them from setting up a permanent protest camp. Those who have been on the front lines for days said they would not back down until the military chiefs step aside.

The protests started without evident organization, much like those that began in January and mushroomed into an 18-day national revolt. The first to take to the streets were for the most part young and liberal activists. When images of a sustained and violent confrontation with police spread online and on television, throngs joined in outrage. In both cases, political leaders opposed to the government appeared to play, at best, marginal roles in the demonstrations.

“It’s either us or them,” protester Ahmed Aggour, 23, said in Tahrir Square on Monday, looking dazed after being shot in the face with a rubber bullet. “Most probably it’s going to be us. We have conviction. They don’t.”

Activists predicted a protracted battle, noting that the generals who replaced Mubarak at the country’s helm also have much at stake.

“To the military leaders, Mubarak’s mistake is he gave concessions to the protesters,” said Hossam Bahgat, a prominent human rights activist in Cairo. “We’re seeing a wholehearted attack not just on the protesters, but on civic society and the media.”

Just outside Tahrir Square, some Egyptians said they were afraid that the confrontation would plunge the country into a deeper crisis.

Shaman Houshi, 22, who works at a bakery, said business has ceased since the new wave of protests began Friday. She said she supported the initial revolution, but not the ongoing effort to unseat the military chiefs.

“Now the people don’t know what they are trying to do,” she said.

Down the street, merchant Adel Mortde, 30, said he hopes the military clears the square — even if it takes a bloodbath.

“I didn’t like Hosni Mubarak,” Mortde said. “But I hate chaos.”

Special correspondent Ingy Hassieb contributed to this report.