By Sunday afternoon, about 580 foreigners had been evacuated from Ndjamena and taken to neighbouring Gabon, French military officials said.
Role of France
France, the former colonial ruler of the central African nation, has reaffirmed its support for Deby's government and offered to help him leave the country, a request he has so far rejected.
The anti-government fighters in turn accuse France of continuing to prop up Deby.
"Instead of evacuating him and rapidly establishing relations with us, France persists in backing Deby," Timane Erdimi, a rebel leader, said in an interview with the French newspaper Journal du Dimanche
Foreign and local residents said heavy weapons and machine-gun fire erupted before dawn on Sunday near the presidential palace to the west of Ndjamena, located on the banks of the Chari river.
Smoke was also seen rising into the sky.
The main market and the national radio station were destroyed by fires and looting, witnesses told the AFP news agency.
While no detailed casualty figures were immediately available, Radio France International quoted the medical charity Medecins Sans Frontieres (MSF, or Doctors Without Borders) as estimating that several hundred people had been injured in last two days of confused street fighting.
One resident told Al Jazeera that he had seen bodies in the streets.
"And I saw cars burning, during the night and all day ... there was heavy shelling over our heads," Fayez el-Fadl, a restaurant owner in Ndjamena, said.
"A lot of people are still stuck in their homes," Haru Mutasa, Al Jazeera's correspondent in Chad, reported.
"They're too scared to leave their houses in case they are caught in the crossfire."
France Info radio reported two French soldiers had been slightly hurt while protecting French and other foreign nationals.
"The city is cut into two - the rebels occupy the west, and the government forces the east," a Reuters reporter in the capital said.
"People are frightened that this fighting will go on."
About 400 people had fled across the western border into Cameroon, according to the UN refugee agency.
Battle for Adre
Rebel fighters were also reported to have surrounded the town of Adre, on the border with Sudan's Darfur region.
"Adre is cut from the rest of the governmental forces. It has been besieged by our forces," Ahmat Soubiane, leader of the Front for the Salvation of the Republic (FSR), said in a statement.
President Idriss Deby seized power in a Libyan-backed coup in 1990
He went on to win the Chad's first two
multi-party elections in 1996 and 2001
A ceasefire signed between Deby and four rebel groups in October recently collapsed
The largest rebel group, the Union of Forces for Democracy and Development, is led by a former minister who accuses Deby of corruption
Click here for more on Chad's spiral into conflict
Based in southeastern Chad, the FSR is not part of the three-group rebel alliance that entered the capital on Saturday.
But Soubiane said his forces were "acting in co-ordination with the rebels in Ndjamena".
"All our organisations have a single goal - the departure of Idriss Deby," he said.
Later, Abderaman Koulamallah, a spokesman for the rebels, said they had overwhelmed a government garrison and seized the town.
But officials said the town was still under the government's control.
"There are many of them, but we are resisting," General Abadi Sair, a government military commander, was quoted by the AFP news agency as saying.
Chadian officials have also blamed Sudan for the fighting, saying that Sudanese forces had backed the rebels in their assault on Adre.
Speaking on Radio France Internationale, Chad's General Mahamat Ali Abdallah Nassour, a military commander, said Sudan's troops attacked Adre from the ground and air, fighting alongside the rebels.
He called it "a declaration of war" by Sudan.
Sudan denies any involvement in the fighting in Chad.