Mr Ouattara's forces have made sweeping gains in the past week but have failed so far to defeat Mr Gbagbo in Abidjan.
Residents say they fear for their safety amid clashes at the presidential palace, TV station and other districts.
France, the UN and US have repeated demands for Mr Gbagbo to stand aside.
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.'He is at home'
Abidjan residents contacting the BBC have said they are afraid to go out of their homes.
One citizen, Maste, told the BBC: "We have been hearing loud gun sounds coming from everywhere and every corner."
Fighting was reported in the Plateau, Agban and Cocody districts, among others. Smoke rose from the area near the presidential palace. The gunfire appeared to lessen in late afternoon and into the evening.
The number of casualties is rising. Mego Terzian, of Doctors Without Borders, told Reuters it had treated at least 80 people in the past two days, many with gunshot wounds, adding: "The majority are young men, though we can not tell whether they are combatants or civilians."