A day after punching deep into the city centre, forces led by the elite counter-terrorism service (CTS) inched towards the governmental compound in Ramadi, the capital of Iraq's vast Anbar province.
"The anti-terrorism troops are now poised to break into the Hoz area where the governmental compound is located," a brigadier general in the force told AFP.
Anbar officials said around 50 families managed to slip away during the fighting around the government compound.
He said thew were mostly children, women and elderly men who raised white flags as they approached the security forces.
Another council official said a family of four were executed by IS on Monday as they tried to leave.
Ibrahim al-Fahdawi, who heads the security committee in Khaldiya, east of Ramadi, said the families numbered around 50.
They were fed and buses were being prepared to take them to a camp away from the frontlines, he said.
Government forces, which have been supported by daily air strikes from the US-led coalition, had to move carefully through the devastated city, whose deserted streets were littered with rubble and shrapnel.
Retreating IS fighters usually booby-trap their abandoned positions, plant roadside bombs and move in tunnels which can also be trapped with huge explosive charges.
Iraqi forces clearing residential neighbourhoods in Ramadi were finding huge amounts of ammunition and explosives, including rockets made from gas canisters.