Sustained bursts of fire rattled lasted for two hours while anti-government demonstrators tried to keep the square.
Egypt's health minister says five people have died in fighting between pro- and anti-government protesters, which began on Wednesday.
The protesters are demanding that President Hosni Mubarak, who has ruled for 30 years, step down immediately.
"Most of the casualties were the result of stone throwing and attacks with metal rods and sticks. At dawn today there were gunshots," Health Minister Ahmed Samih Farid told state television.
"The real casualties taken to hospital were 836, of which 86 are still in hospital and there are five dead."
The US has urged all Americans in Egypt to leave "immediately".
Mr Mubarak has said that he will serve out his current presidential term, which ends in September, but will not run for re-election.
On Wednesday, groups fought pitched battles in Cairo, in the worst violence in 10 days of protests.
The unrest has left about 300 people dead across the country over more than a week, according to UN estimates.
Cairo's Tahrir Square has been the main focus of the protests.
Reports early on Thursday suggest that supporters of President Mubarak have been firing on people in the square from the October Bridge, with unknown numbers wounded.
Speaking to the BBC early on Thursday, Mona Seif, a protester, described the atmosphere as "very tense".
"Every couple of minutes we hear a sequence of gunshots, and it's only on one side of the square, which is the one close to Cairo Museum. This is where the clashes have been going on for more than five hours now, completely non-stop," she said.
Wednesday's violence began when thousands of supporters of President Mubarak surged into the square.
One anti-government protester told the BBC that the pro-Mubarak activists had initiated the fighting.
"They started throwing stones at us," the man, named as Zaccaria, said. "Then some of us started throwing stones at them and then we chased them out of the square. They returned once again with the horses and the whips and the thugs."
Opposition supporters say many in the pro-government camp were paid by the authorities to demonstrate, and allowed into the square by the troops surrounding it.
The two sides pelted each other with stones in running battles lasting for hours.
Egyptian troops refused to intervene, but fired into the air to try to disperse people. On Wednesday, they called for demonstrators to return to their homes, a call which was reiterated by Vice-President Omar Suleiman.
Clashes were also reported in Egypt's second city, Alexandria.
The US State Department has urged all Americans in Egypt who wish to leave to go to airports "immediately", adding that delay was "not advisable".
"Additional US government flights after Thursday are unlikely," the State Department added.
US Secretary of State Hillary Clinton told Egyptian Vice President Omar Suleiman in a phone call on Wednesday that the clashes were a "shocking development" after days of peaceful protests.
She also "underscored the important role that the Egyptian Armed Forces have played in exercising restraint in the face of peaceful demonstrations", the state department said in a statement.
End Quote Mark Mardell BBC North America editor
The White House, as much as the pro-democracy protesters, is demanding 'Mubarak must go'”
The BBC's Rupert Wingfield-Hayes says he was handcuffed, blindfolded and interrogated by Egyptian secret police, before being released after three hours.
He says that the authorities' attitude appears to be hardening and the ruling elite are fighting back.
The violence drew condemnation from British Prime Minister David Cameron.
"If it turns out that the regime in any way has sponsored or tolerated this violence, that is completely unacceptable," he said after meeting UN Secretary General Ban Ki-moon in London.
Mr Ban said: "Any attack against the peaceful demonstrators is unacceptable and I strongly condemn it."
If Mr Mubarak does not step down, demonstrators have planned to march on the presidential palace on Friday.
Meanwhile, internet services were returning to the country, having been cut off for days by the government.