BAGHDAD — For the first time, Iraqi forces engaged Islamic State fighters within the city center of Ramadi on Tuesday, reaching the edge of the inner government district in an attempt to seize the critical western provincial capital after months of approach and maneuvering, officials said.

“We went into the center of Ramadi from different axes, and we started clearing residential areas,” Gen. Sabah al-Numani, a spokesman for the army counterterrorism unit in charge of the offensive, said in a statement Tuesday. He predicted that “the city will be cleared within the coming 72 hours.”

Six hundred to 1,000 Islamic State fighters were said to have been in Ramadi when the overall offensive began two weeks ago, but several hundred of them have been killed in fighting and airstrikes since then, according to Iraqi and Western officials.

Those remaining did not appear to be giving up easily. Iraqi forces, including a mix of soldiers and policemen along with a contingent of Sunni tribal fighters, faced heavy fire and were assaulted by car bombs, Iraqi officials said. And fighters for the Islamic State destroyed three bridges over the Euphrates River to slow the security forces’ advance, according to Gen. Ahmed al-Belawi, the leader of a battalion of Sunni tribal fighters. The force crossed on Tuesday using portable bridges supplied by American forces, officials said.