There are growing fears for the fate of civilians trapped in Manbij, formerly a key stop along IS's supply route from neighbouring Turkey into its self-styled Islamic "caliphate" in Syria.
The US-backed Syrian Democratic Forces (SDF) penetrated the town one month ago but have since been hindered by a bloody IS counter-offensive amid concerns about civilians.
On Saturday, IS jihadists appeared to ignore a 48-hour ultimatum to leave issued by the Manbij Military Council, a key SDF member.
"The 48-hour period is over, and there will be no more opportunities like this one for Daesh (IS)," a commander from the council told AFP on condition of anonymity.
IS has "not responded" to the SDF's offer and instead "attacked our positions", he said.
The ultimatum came after at least 56 civilians, including children, were reportedly killed Tuesday in US-led coalition air strikes near Manbij.
The commander pledged that his forces would "intensify our attacks on their remaining positions".
The council's spokesman, Sherfan Darwish, stressed in an online statement on Saturday that SDF forces in Manbij "are committed to securing safe passage to the best of their ability for any civilian able to flee Daesh (IS)'s brutality".
Thousands of civilians have already fled Manbij, which is in Syria's northern Aleppo province.
More than half of Syria's population has been displaced since the conflict erupted in 2011, and at least 280,000 people have been killed.
'Fight like we haven't seen'
The Syrian Observatory for Human Rights said SDF forces were advancing in Manbij, moving steadily north from districts they already control in the west and south.
An SDF field commander inside Manbij told AFP on Saturday that the main clashes were "near the security quarter in the centre of the town".
The civilian deaths in Tuesday's raids sparked an intense backlash from activists and rights groups, and a call from a prominent Syrian opposition body for the coalition to halt its air campaign until a thorough investigation is completed.
The coalition has said it is investigating the reports of civilian fatalities in the town of Al-Tukhar, 14 kilometres from Manbij.
Bombing raids have meanwhile continued unabated, with the Pentagon reporting nine strikes near Manbij on Friday.
Coalition spokesman Colonel Chris Garver said on Friday that IS was mounting an exceptionally tough fightback.
The battle has intensified as SDF units move deeper into the town, he said, "which is sort of different than what we saw in Ramadi and what we saw in Fallujah", two Iraqi cities from which jihadists were ousted this year.
"It's a fight like we haven't seen before."
IS was using residents "as human shields and as bait" in order to draw SDF fire towards civilians, Garver added.
Observatory head Rami Abdel Rahman accused IS of "pushing children towards the frontlines" as it tried to defend its positions.