The pro-GNA forces have been engaged in a military operation since May 12 to retake the coastal city located 450 kilometres (280 miles) east of Tripoli.
Sirte has been a stronghold of the jihadist group since June last year.
On Monday, the loyalists said they had seized an area of guest houses close to the Ouagadougou conference centre, the complex from the era of slain dictator Moamer Kadhafi where the jihadists have their headquarters.
The field hospital of the pro-GNA forces near Sirte posted on Facebook that four people among its ranks were killed and 32 wounded on Monday.
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Pro-GNA forces entered Sirte in June, but their progress slowed as the jihadists hit back with sniper fire, suicide attacks and car bombings.
Since Thursday pro-government forces have been battling to reach the conference centre.
"Our forces have targeted Daesh snipers and their mines," the statement said, using an Arabic acronym for IS.
The guest houses are east of the Al-Dollar district, which the pro-GNA forces took last week and where VIP guests attending events at the Ouagadougou complex used to be accommodated.
On Sunday, the loyalist forces said they would soon begin a final assault to retake Sirte from IS.
The United States has since last Monday been carrying out air raids on IS positions in the city at the GNA's request.