Hundreds of Islamists loyal to deposed president Mohamed Morsi clashed with security forces, hours after a US envoy urged Egypt's army-backed leaders to end violence.
US envoy Bill Burns also urged the army to avoid "politically motivated arrests" amid growing international unease at the crackdown on Morsi's Muslim Brotherhood.
He was speaking after talks with the new leadership, including the army-appointed interim premier Hazem al-Beblawi, who is expected to announce a new cabinet Tuesday or Wednesday.
But the Brotherhood has refused to take any role in the new government and its supporters are still out on the street calling for Morsi's reinstatement.
Late on Monday hundreds of demonstrators cut off the October 6 bridge across the Nile in the heart of Cairo.
Security forces fired tear gas to drive them back, an AFP correspondent reported. The protesters responded by hurling rocks at the security forces, who responded with fresh volleys of tear gas.
The demonstrators had turned out in their thousands in Cairo after the Ramadan iftar meal to demand Morsi's return.
The clashes were the first in the Egyptian capital since dozens of pro-Morsi demonstrators were shot dead outside an elite military headquarters the previous Monday.
Hours earlier, Under Secretary of State Bill Burns called for dialogue to replace the violence.
"The first priority must be to end violence and incitement, prevent retribution, and begin a serious and substantive dialogue among all sides and all political parties," he said.
He was speaking after meeting the general behind the coup, army chief Abdel Fattah al-Sisi, as well as the military-appointed president Adly Mansour and Beblawi.
Burns' visit is the first to Egypt by a senior US official since the military toppled Morsi in a popularly backed coup on July 3.
Egypt's new leaders are pushing ahead with a transition plan for an interim government and fresh elections, despite the continuing protests by Morsi loyalists.
A State Department spokeswoman confirmed that Burns had not met any Brotherhood officials.
And Tamarod, the movement that spearheaded the grassroots campaign against Morsi, said it had turned down a chance to meet the US envoy.
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"We rejected the invitation... because the United States did not stand with the Egyptian people from the beginning," Islam Hammam, one of the group's organisers, told AFP.
Burns' visit comes as the authorities tighten the screws on Morsi's backers, freezing the assets of 14 top Islamists, and with Egypt rocked by a wave of deadly attacks, notably in the Sinai.
Three factory workers were killed in a rocket-propelled grenade attack in the restive peninsula, medics said.
Coptic Christians have also been killed in the Sinai, including a priest, as part of a what an Egyptian rights group said was a surge in sectarian violence around the country since the Islamist president's overthrow.
In Paris, UN chief Ban Ki-moon warned against "revenge" and "retribution", and denounced the arrests of Brotherhood officials.
International concern is mounting over the fate of Morsi, who has been in custody since the coup. He was quizzed by prosecutors on Sunday over complaints of possible criminal offences.
Washington has refrained from saying he was the victim of a coup, which would legally require a freeze on some $1.5 billion in desperately needed US military and economic assistance to Cairo.
Burns declined to comment on Morsi, saying only: "We've called on the military to avoid any politically motivated arrests."
Several leading Republican lawmakers have already called for US aid to Egypt to be cut because the army's removal of Morsi.
On Monday Kenneth Roth, executive director of Human Rights Watch, urged Egyptians to be wary of recent developments. "Dictatorship from the right can wear beards or uniforms (and a liberal-endorsed one is no better)," he tweeted.
Morsi's supporters say his overthrow was an affront to democracy.
Before the clashes on the October 6 bridge, tens of thousands of pro-Morsi protesters gathered outside Cairo's Rabaa al-Adawiya mosque, where they have rallied for the past two weeks.
Egypt's military-appointed leadership has been trying to install a new cabinet to oversee the transition to parliamentary and presidential elections.
Burns said Washington hoped that Egypt would swiftly return to a democratically elected civilian government.
Brotherhood officials have vowed to continue "peaceful" protests until the reinstatement of Morsi, who interim leaders say is being held in a "safe place, for his own safety".
During his single year of turbulent rule, Morsi was accused of concentrating power in Brotherhood hands, sending the economy into freefall and failing to protect minorities.