Backed by weeks of US air strikes, the pro-GNA forces have recaptured nearly all of what had been the jihadists' main stronghold in North Africa.
The city's fall would be a huge setback to IS's efforts to expand its self-proclaimed "caliphate" beyond Syria and Iraq where the jihadists have also suffered losses.
"The fighting has begun. We are attacking the last Daesh positions in district three" where the jihadists are cornered, a GNA fighter told AFP, using an Arabic acronym for IS.
"Our forces are advancing inside the areas where Daesh is, in district three, and so far have taken control of" two banks and a hotel, the media centre said.
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It also said they had thwarted an attempted suicide bombing.
Seven members of the GNA forces were killed and 30 wounded in clashes with IS on Saturday, it added.
Earlier, an AFP journalist saw ambulances leaving Sirte -- hometown of dead dictator Moamer Kadhafi -- for Misrata to the west where the wounded are treated.
It is not yet known how many IS militants have been killed, but the GNA media centre said the bodies of 10 jihadists had been found in a school in district one, which was being combed after being retaken on Monday.
The forces loyal to the UN-backed GNA had said last weekend they were preparing to "liberate" the entire city after seizing several IS positions, including its headquarters.
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