Losing Raqa would be only the latest in a series of crushing defeats for the extremist group, which once controlled large swathes of territory spanning the border between Syria and Iraq.
Captured by IS in 2014, the northern city was the de facto Syrian capital of the jihadists' self-styled "caliphate" until the US-backed assault by the Syrian Democratic Forces, an alliance of Kurdish and Arab fighters.
The militia has captured around 90 percent of Raqa since entering the city in June, after months of fighting to encircle it.
They are now advancing on IS-held districts from two fronts in the city's north and east, commander Rojda Felat, who heads the "Wrath of the Euphrates" campaign, said Sunday.
She said fighting was still fierce along the front line, with IS using snipers, suicide bombers and reinforced positions in tunnels to hold up the SDF advance.
- Handful of positions left -
The jihadists still hold Raqa's national hospital, the nearby football stadium and surrounding residential neighbourhoods, including the infamous Al-Naim roundabout, where IS staged public beheadings and crucifixions.
"There were intermittent clashes today around the hospital and the stadium," the head of the SDF's press centre Mustefa Bali said Monday.
"There was no notable or tangible advance, but shelling and sniping operations are ongoing."
SDF fighters have surrounded the hospital ahead of a push to seize Al-Naim, said Ali Sher, a field commander with the Kurdish People's Protection Units (YPG), which make up the bulk of the SDF.
IS is believed to be holding civilians as human shields in the hospital, complicating efforts to capture the position.
Colonel Ryan Dillon, spokesman for the US-led coalition backing the SDF's assault, said IS was using the hospital as a military base and it was "heavily fortified".
He said coalition special forces advisers could accompany the SDF in a push for the facility, but there would not be "full, tactical, coalition units assaulting the hospital".
Washington's envoy to the coalition, Brett McGurk, said Monday that its forces had carried out 75 air strikes within 48 hours to prepare for an SDF assault on the jihadists' remaining holdouts.
"SDF fighters advancing room-by-room through city center. 17 city blocks cleared yesterday," he tweeted.
Tens of thousands of civilians have fled Raqa city and the surrounding area since the SDF began its offensive, but many others have remained trapped inside during the heavy fighting.
Laila, 32, escaped on Sunday from a building near the stadium.
A mother-of-three, and pregnant with her fourth child, she feared her missing husband had been killed in shelling.
- 'Screaming under the rubble' -
She described utter terror as she sheltered with her children during ferocious air strikes and mortar fire that collapsed buildings around them.
"Those who were still alive were screaming under the rubble but no one dared pull them out because there was so much shelling," she said.
"We lived for three months in the bathroom. My son kept telling me, 'I just want to see the sun, I just want to see the sun.' Today he saw it for the first time."
The loss of Raqa city would leave IS with just a handful of positions in Syria and Iraq.
The group was forced out of Iraq's Mosul in July and last week was driven from Hawija, meaning it holds just a sliver of territory in the Euphrates Valley near the border with Syria.
The SDF has already begun a new campaign against the jihadists to retake territory they hold in the eastern province of Deir Ezzor, which neighbours Raqa and sits on the border with Iraq.
They are fighting on the eastern side of the Euphrates River that slices diagonally across Deir Ezzor.
Syria's army is fighting a separate Russian-backed campaign largely on the western bank of the river, and last month broke an IS siege of nearly three years on parts of Deir Ezzor city.
Turkey-backed Syrian rebels are also preparing for an operation to oust jihadists from the northwestern province of Idlib.
On Monday, Turkey's army said it had launched a reconnaissance mission in Idlib days after President Recep Tayyip Erdogan announced an incursion to oust Al-Qaeda's former Syrian affiliate from the area.