The SDF said it had seized al-Mishlab district in the far east of Raqqa on Friday and al-Sabahia district in the west.
The capture of Romaniah yesterday by the US -backed, Kurdish-led Syrian Democratic Forces marks the second district to fall out of ISIS' control after a wide offensive on Raqqa, among the first cities ISIS captured in 2014 and home to some of its most prominent leaders.
The UK-based Syrian Observatory for Human Rights said on Monday that at least 24 civilians have been killed in air strikes and fighting in the past 48 hours.
The ground offensive against al-Raqqa, the main IS stronghold in Syria, began last week, under the command of the Kurdish-led Syrian Democratic Forces.
Speaking on condition of anonymity, the source said the Kurdish-led forces are expecting to capture Raqqa within two months.
The US-led coalition estimates that Raqqa, which Islamic State seized from Syrian rebels in 2014, is defended by 3,000 to 4,000 militants.
Captured by the jihadists in 2014, Raqa has become synonymous with IS atrocities including beheadings and public displays of bodies, and also emerged as a hub for planning attacks overseas.More news: Man on life support after Milwaukee lakefront shooting
The report of Baghdadi's death comes amid a military push by US -backed Syrian forces into Raqqa, the jihadist group's last remaining stronghold in the region.
Bizarrely, the New York Times reported this claim under the headline: "U.S".
The fighters have also entered Raqqa's western neighborhood of Sabahiya and the industrial district in the east.
The city has always been seen as a prize by multiple parties to the Syrian conflict, including the government, Russia, Turkey and the US-led coalition bombing IS since 2014.
An estimated 300,000 civilians were believed to have been living under ISIL rule in Raqqa, including 80,000 displaced from other parts of Syria.
But thousands have fled in recent months, and the United Nations humanitarian office estimates about 160,000 people remain in the city. Even with assistance from USA military advisors and air support, it has taken Iraqi forces nine months of prolonged combat to reach Mosul's Old City.
Back in December, US Petagon spokesman, Peter Cook, told CNN that the coalition was determined to "deliver him the justice he deserves".