Troops supporting the UN-recognised president of Ivory Coast, Alassane Ouattara, are taking part in a final push to oust his rival, Laurent Gbagbo, who refuses to give up the presidency.
In the main city, Abidjan, pro-Ouattara forces have launched an assault on the fortified presidential residence.
French troops have taken about 500 foreigners to a military camp amid increasing insecurity in the city.
Mr Ouattara's supporters began a fresh offensive on Monday.
Mr Ouattara was internationally recognised as president last year, after the electoral commission declared him the winner of a November run-off vote, but Mr Gbagbo also claimed victory.
Air border reopens
There is fierce fighting in the area around the presidential residence in Abidjan. "I can still hear heavy gunfire and loud thud of mortar fire," one resident told the BBC.
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I've just had a phone call with Charles Ble Goude, the second most important man in Laurent Gbagbo's team in Ivory Coast. He is the leader of the Young Patriots.
He told me that he had a message for the outside world. He wants them to know that he wants to avoid a civil war and bloodshed in Abidjan. He was asking Mr Ouatarra's people to have a dialogue and to negotiate in order to save lives before, he said, it's too late.
He said it is not a matter of controlling or not controlling the country. It is a matter of principle. That is why, he said, he is calling for a ceasefire because it is never too late, he said.
He said that Laurent Gbagbo was not surrounded at his residency or in any danger. He said the forces of Laurent Gbagbo controlled the city and he was not yet calling his Young Patriots on to the street. He said the restraint was because: "I don't want people to push me to take them on to the streets. People must be cautious of their responsibilities."
A military source close to the Gbagbo camp told Reuters news agency that pro-Ouattara forces were attacking the compound, adding that the Republican Guard and militiamen were fighting back.
A plume of smoke has been seen rising from the residence, which is located on a peninsula in Abidjan's lagoon.
BBC correspondents also say fighting is very intense around the main police camp in Agban and in the area of Le Plateau, not far from the presidential palace.
Mr Gbagbo has not been seen in public for weeks. It is not clear whether he is in the compound.
Mr Gbagbo's adviser in Europe, Toussaint Alain, told Associated Press: "President Gbagbo has not fled the country. He is on Ivory Coast territory, in a safe place from where he is organising and leading the resistance."
French forces say they have taken about 500 foreigners, including 150 French nationals, to a military camp after they were threatened by looters in Abidjan.
"There is a security vacuum and that has opened the way for looters to roam the streets," French military spokesman Thierry Burkhard told Reuters.
A Swedish woman working for the UN in Abidjan was killed by gunfire at her home late on Thursday.
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Ivory Coast: Battle for power
- World's largest cocoa producer
- Once haven of peace and prosperity in West Africa
- Alassane Ouattara recognised as president-elect in 2010
- International sanctions imposed to force Laurent Gbagbo to go
- 473 killed, one million fled since disputed election
- 9,000 UN peacekeepers monitor 2003 ceasefire
- In pictures: showdown looms
- Profile: Laurent Gbagbo
- Profile: Alassane Ouattara
- Eyewitnesses: Looting and shooting
The AFP news agency on Friday reported French officials as saying a teacher of French nationality had been shot dead in Yamoussoukro. Ivory Coast's capital is in the hands of Ouattara forces but it was not known if the shooting was linked to the political unrest.
Mr Ouattara's government said Ivory Coast's land, sea and air borders had been closed until further notice, although later on Friday it said that the air border had now reopened.
UN and French peacekeepers have taken control of Abidjan's international airport.
Mr Ouattara's government has declared there will be a curfew from 2100 GMT to 0600 GMT in Abidjan until Sunday.
Mr Gbagbo's army has put up almost no resistance since the start of the offensive by Mr Ouattara's supporters, who are now believed to control about 80% of the country.
Mr Ouattara's forces have made lightning advances since Monday, moving out from their base in the north. On Wednesday, they captured Yamoussoukro and the key port of San Pedro.
Many senior military officers, including army chief Gen Phillippe Mangou, as well as thousands of soldiers and police, have abandoned Mr Gbagbo.
However, he retains the support of the Republican Guard, special forces and armed militias known as the Young Patriots
BBC News - Ivory Coast: Gbagbo and Ouattara in Abidjan battle
A spokeswoman for Mr Ouattara, Anna Ouloto, said: "I don't think Laurent Gbagbo is capable of resisting for much longer with all the defections in his ranks... he is condemned to be removed."
On Friday the West African bloc Ecowas again urged Mr Gbagbo to give up power "to end the suffering of his country".
The call to step aside was repeated by the office of French President Nicolas Sarkozy and by UN Secretary General Ban Ki-moon.
The UN mission in Ivory Coast (UNOCI) said the UN's special representative there had spoken to Mr Gbagbo's allies and was ready to "facilitate his departure if that is his wish".
The UN, which helped organise last year's election, certified Mr Ouattara's win as legitimate. But Mr Gbagbo claimed the presidency after the Constitutional Council overturned Mr Ouattara's victory.
Sanctions and a halt to cocoa exports in what is the world's biggest producer of cocoa beans have brought West Africa's second-biggest economy to its knees, with banks closed for more than a month.
An armed rebellion in 2002 split the nation in two - a division the elections were meant to heal.
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