The Islamic State has rejected an offer for its fighters to withdraw safely from the Syrian city of Manbij, and clashes with US-backed forces continued after the offer expired.

The militant group did propose an alternative, which would allow the safe passage of critically ill civilians to areas controlled by US-backed forces, in exchange for allowing wounded IS fighters to leave the city.

The Syrian Democratic Forces (SDF) alliance — made up of Arab and Kurdish fighters and backed by the US — made rapid advances against IS last month after launching an aggressive campaign to flush them out of the area along the strategically important Turkish border area, which militants had used for years to traffic weapons and fighters.

The SDF quickly surrounded Manbij — a city in the Aleppo governorate — but the fight to actually recapture the city has been more difficult.

On Thursday, the SDF-allied Manbij Military Council put an offer on the table – IS militants would be given safe passage out of the city with light weapons, without a fight, if they left within 48-hours.

But IS wasn't interested.

"The deadline is approaching, time is almost up," the Council's spokesman Sharfan Darwish said on Saturday. "As far as we're concerned, the situation has not changed," adding that there had been no response from IS to the proposal.

Darwish did not say whether the SDF would accept the IS proposal on wounded fighters and sick civilians. He said IS had been using civilians as "human shields".

The British-based Syrian Observatory for Human Rights said clashes continued inside Manbij, and that the US-led coalition fighting IS was bombing parts of the city and nearby areas.

Activists say at least 56 civilians were killed in US-led coalition airstrikes on Manbij this week. The opposition Syrian National Coalition called for a suspension of the air strikes while the incidents are investigated.