Witnesses have reported hearing intense gunfire near Mr Gbagbo’s residence, while Mr Ouattara’s supporters say they have taken control of state television.
His government earlier closed Ivory Coast’s borders and declared a curfew.
Mr Gbagbo has refused to relinquish the presidency since November’s election.
But the national army has put up almost no resistance since Mr Ouattara’s supporters launched an offensive on Monday.
Pro-Ouattara forces reportedly now control about 80% of the country.
As the battle for control of the country appeared to reach a climax, gunfire was heard around several strategic buildings in Abidjan.
Heavy fighting was reported close to the headquarters of RTI state TV and Mr Gbagbo’s residence, both in the northern district of Cocody.
“The gunfire has been intense and they’re shooting in four or five directions at a time. There’s a lot of people,” a resident told the AFP news agency. “It looks like a final assault.”
A spokesman for Mr Ouattara’s government, Patrick Achi, said the former president had so far shown no signs of giving up.
Mr Achi said Ouattara loyalists had taken control of RTI. This could not be confirmed, but the channel went off-air late on Thursday.
A spokesman for Mr Gbagbo said his forces were still fighting at the TV station.
Mr Gbagbo has not been seen in public for weeks. His residence is mainly protected by members of the elite presidential guard, and is located on a peninsula in Abidjan’s lagoon.
“Gbagbo is in his house. I’m certain. He hasn’t gone anywhere,” said Mr Achi.
Earlier, Mr Ouattara’s government said Ivory Coast’s land, sea and air borders had been closed until further notice. It also declared that there would be a curfew from 2100 GMT to 0600 GMT in Abidjan until Sunday.
And after looting was reported in several parts of the city, UN and French peacekeepers took control of Abidjan’s international airport.
The BBC’s John James in Bouake says growing panic seems to be setting in among Mr Gbagbo supporters, especially following the decision of the head of the army, Gen Phillippe Mangou, to seek refuge with his wife and five children at the home of the South African ambassador.
On Thursday evening, Mr Ouattara’s television channel featured several high-level military officers pledging allegiance to his government. A source also told the BBC that the head of the gendarmerie, Edouard Kassarate, had defected.
The head of the UN mission, Choi Young-jin said that as many as 50,000 soldiers, police and gendarmes had abandoned Mr Gbagbo, with only the Republican Guard and special forces personnel remaining loyal.
“[My troops] have come to restore democracy and ensure respect of the vote by the people,” Mr Ouattara said in an address. “To all those who are still hesitating, whether you are generals, superior officers, officers, sub officers, rank-and-file… there is still time to join your brothers-in-arms.”
Western diplomats say it is only a matter of time now before Mr Gbagbo flees or is captured, our correspondent says.
Mr Ouattara’s government is giving assurances that the outgoing president will not be harmed, he adds. They say, instead, that Mr Gbagbo will be made available to the International Criminal Court.
Earlier, UN Secretary General Ban Ki-moon again demanded that Mr Gbagbo immediately cede power to Mr Ouattara “to enable the full transition of state institutions to the legitimate authorities”.
The US Assistant Secretary of State for African Affairs, Johnnie Carson, urged both sides to exercise restraint and protect civilians. Both Mr Gbagbo and his wife would be held accountable if significant violence broke out, he added.
Mr Ouattara was internationally recognised as president last year, after the electoral commission declared him winner of the November run-off vote.
The UN, which helped organise the vote, certified it as legitimate. However, Mr Gbagbo claimed victory after the Constitutional Council overturned Mr Ouattara’s win.
The forces supporting Mr Ouattara have made lightning advances since Monday, moving out from their base in the northern half of the country.
On Wednesday, his fighters captured Ivory Coast’s capital, Yamoussoukro, and the key port of San Pedro. Mr Gbagbo’s hometown of Gagnoa also fell.
Since the crisis began in December, one million people have fled the violence – mostly from Abidjan – and at least 473 people have been killed, according to the UN.
Sanctions and a boycott on 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.