At least 180 civilians in areas controlled by IS have managed to flee, with the Syrian Observatory for Human Rights putting that number in the hundreds.
The Syrian Democratic Forces (SDF), who are Kurdish led, have been one of the most effective groups fighting IS in the war-torn country.
Their offence to capture Raqqa was launched on 6 June, and they have already seized control of several areas.
Reports suggest the fighting is concentrated in the southwestern neighbourhood of Yarmouk, with SDF fighters taking positions near a centuries-old mosque in the city’s centre.
The SDF says 11 IS militants have been killed in the clashes since Sunday, with the IS-linked Amaq news agency claiming 14 SDF fighters died in Sunday’s fighting alone.
It comes a week after Iraqi forces declared victory against IS in the northern Iraqi city of Mosul, the largest the extremists have held.
Although the loss of Raqqa would deal a major blow to IS, the group still holds wide areas of Deir al Zor – an eastern province which borders Iraq.
This area could become the backdrop to the next phase of the war on Islamic State, with government forces backed by Russia and Iran already approaching from the west and south.
Pro-government forces reached the edges of the Bashari mountain range on Monday after seizing a natural gas field over the weekend.
Meanwhile, EU foreign ministers have agreed a fresh round of sanctions against the Syrian regime of Bashar al Assad over its use of chemical weapons against its own people.
Foreign Secretary Boris Johnson said 16 named military officers and scientists would be subjected to travel bans and asset freezes after a meeting with his counterparts in Brussels.