Iraqi forces are attempting to move further into the centre of Ramadi, on the second day of an assault to drive Islamic State militants from the city.
Security officials say troops and Sunni tribal fighters have taken control of several districts and are advancing towards the main government complex.
The army's chief-of-staff expects the up to 300 militants inside the city centre to be dislodged within days.
But there is concern for the civilians they have reportedly taken prisoner.
Sources in Ramadi said on Tuesday that the jihadists had carried out raids and mass arrests in an attempt to prevent an uprising in support of the government offensive by the thousands of people living in districts under their control.
Ramadi, a predominantly Sunni Arab city about 90km (55 miles) west of Baghdad, was captured by IS in May in an embarrassing defeat for the army.
Retaking it would be a "huge morale and strategic boost" for the Iraqi security forces, former national security adviser Mowaffaq al-Rubaie told the BBC.
Elite Counter-Terrorism Service (CTS) forces launched the assault on central Ramadi at dawn on Tuesday, with the support of soldiers, police, Sunni tribesmen opposed to IS, and US-led coalition air strikes.
By the afternoon, government forces had retaken the al-Thubat and al-Aramil districts, and entered nearby al-Malaab and Bakir, security sources told the BBC.
Floating bridges built over River Euphrates, which flows along the north and west of the city centre, also enabled troops to enter directly the al-Haouz district, near the government complex.
On Wednesday morning, a spokesman for the governor of Anbar province, of which Ramadi is the capital, told the BBC that most of the IS militants who were in Ramadi before the government began the operation to retake the city in November had fled or been killed.
"We think there are no more than 300 foreign fighters from Daesh still fighting, probably the suicide bombers. But overall, we think we are very close to liberating the city," Muhannad Haimour said, using a pejorative term for IS based on the acronym of its previous name in Arabic.
"It's a very, very difficult battle, especially with so many booby-traps, explosives and sniper fire against the security forces and tribal fighters."
Mr Haimour said the jihadists had taken many men prisoner and prevented their families from leaving.
See where IS have made territorial gains and losses, Jan-Dec 2015
"It's very difficult to estimate the numbers. We're probably talking about 5,000 people who have been forced to stay by Daesh," he added.
The operation to recapture Ramadi, which began in early November, has made slow progress, mainly because the government has chosen not to use the powerful Shia-dominated paramilitary force that helped it regain the northern city of Tikrit to avoid increasing sectarian tensions.
IS has lost control of several key towns in Iraq to government and Kurdish forces since over-running large swathes of the country's west and north in June 2014 and proclaiming the creation of a "caliphate" that also extended into neighbouring Syria.
What is Islamic State?
A notoriously violent Islamist group which controls large parts of Syria and Iraq. It has declared its territory a caliphate - a state governed in accordance with Islamic law - under its leader Abu Bakr al-Baghdadi.
What does it want?
IS demands allegiance from all Muslims, rejects national borders and seeks to expand its territory. It adheres to its own extreme interpretation of Sunni Islam and regards non-believers as deserving of death.
How strong is IS?
IS projects a powerful image, partly through propaganda and sheer brutality, and is the world's richest insurgent group. It has about 30,000 fighters but is facing daily bombing by a US-led multinational coalition which has vowed to destroy it.