One witness said Ouattara troops have begun an assault on the fortified presidential palace in Abidjan, the BBC reported Friday.
"I can still hear heavy gunfire and loud thud of mortar fire," the witness told the BBC. "And it is coming from the direction of Cocody," where the presidential residence is located.
Gbagbo has not been seen in public for weeks and it is unclear if Gbagbo is in the presidential compound.
Presidential spokesman Abdon Georges Bayeto said Gbagbo "is not going to step down. He's been elected for five years and we are going to put up a fight."
Gbagbo has refused to relinquish the presidency since November's election despite Ouattara being recognized as the winner by the international community. However, Gbagbo claimed victory after the Constitutional Council overturned Ouattara's win.
Witnesses said the national army has offered little resistance since the start of the offensive by Ouattara's supporters earlier this week. Pro-Ouattara forces reportedly control about 80 percent of the country.
High-level supporters of Gbagbo have been defecting to Ouattara, Ivory Coast officials said.
Phillippe Mangou, head of the army and a longtime Gbagbo loyalist, sought refuge in the South African embassy in Abidjan with his wife and children, the BBC reported. A source told the BBC Edouard Kassarate, head of the military police, defected to Ouattara with his forces.
Many top government officials appear to have sent their families to Europe.
Choi Younjin, head of the U.N. mission in Ivory Coast, said as many as 50,000 soldiers, police and gendarmes had abandoned Gbagbo, leaving him with only the Republican Guard and special forces personnel.