Witnesses told the BBC fighters were storming the palace amid reports of "heavy gunfire."
At least 1 million people were displaced from Abidjan and at least 450 were killed since fighting began last year. Gbagbo refuses to step down despite recognition by the international community that Ouattara won the election.
The British Foreign and Commonwealth Office said Friday that it was "gravely concerned" by the level of violence in Abidjan.
"We call on all sides to exercise restraint," the statement read. "Reports of human rights violations must be investigated and those responsible held to account."
The U.N. High Commissioner for Refugees issued reports describing a "mass exodus" from Abidjan and international rights groups said the situation was characteristic of civil war.
U.N. Secretary-General Ban Ki-moon, in a statement issued through his office, called on "former President Laurent Gbagbo to immediately cede power to President Ouattara to enable the full transition of state institutions to the legitimate authorities."