France's prime minister, Francois Fillon, said French representatives were in negotiations with two Ivory Coast generals loyal to Mr. Gbagbo, The New York Times reported.
Gbagbo was in a bunker beneath his residence, Ouattara's ambassador to France said early Tuesday.
"I believe Laurent Gbagbo is alive. I have learned that he is negotiating his surrender," Ally Coulibaly told Radio France Internationale in an interview.
Coulibaly said Gbagbo has been "seeking to negotiate" since Monday.
Ouattara is recognized by the international community as the winner of a fraud-riddled election in November, but a court threw out the results, leaving Gbagbo to maintain he is still president.
In Abidjan, weapons fire was heard Tuesday as Ouattara's forces fought against Gbagbo loyalists, RFI reported.
A Ouattara spokesman said the presidential residence in the district of Cocody had been taken over.
Monday, a joint French-U.N. military operation struck targets at Gbagbo's residence, his offices and two of his military bases, the Times reported.
French forces attacked heavy artillery and armored vehicles at Gbagbo's residence and presidential offices, a military spokesman said. The United Nations said it launched helicopter strikes against Gbagbo's forces at two of his bases to prevent pro-Gbagbo forces from firing on civilians and United Nations personnel.
French officials said in a statement the country's forces were participating in the operation at the request of the United Nations, with the intent of "neutralizing heavy weapons that are used against the civilian population and United Nations personnel in Abidjan." France, a former colonial ruler of Ivory Coast, has more than 1,500 troops stationed in the country.
U.S. President Obama Monday urged Gbagbo to "end his claim to the presidency" and demonstrate respect for Ivory Coast citizens