Smoke and fumes from the ship at the San Diego Naval Base continued to pollute the skyline and air throughout the California city, The San Diego Union Tribune newspaper said in a report on Monday.
In an statement on Monday evening, a Naval Surface Forces spokeswoman said crews have made "significant progress" in the effort to save the ship.
Rear Adm. Philip Sobeck, the commander of Expeditionary Strike Group 3, said that the fire was in the superstructure of the ship and its upper decks and that the ship's forward mast has collapsed.
"There's obviously burn damage all the way through the skin of the ship, and we are assessing that as we kind of go through each compartment," he was quoted as saying in The San Diego Union Tribune newspaper report.
"Right now the priority is to get the fire out so that we can take a complete assessment," Sobeck said, adding that the flames were two decks away from the area in the ship where a million gallons of fuel is stored.
About 400 sailors and fire crews from across the San Diego waterfront are fighting the blaze, and helicopters are doing water drops.
So far, 36 sailors and 23 civilians have been treated mostly for minor injuries related to firefighting.
The cause of the fire is still unclear.
Sobeck was quoted as saying by The San Diego Union-Tribune that the fire may have begun in a cargo area known as the deep V storage area.
There are large storage boxes, called triwall boxes, that are the size of pallets that burned.
USS Bonhomme Richard is a Wasp-class amphibious assault ship home-ported in San Diego and has a crew size of around 1,000.