The United States welcomed the advance on Sirte, the hometown of ousted dictator Moamer Kadhafi, which has also been in the sights of forces of a rival authority in eastern Libya.
The loss of Sirte would amount to a huge setback for IS, which is also faced with battlefield losses in Syria and Iraq.
"The armed forces entered Sirte. They are currently in the centre, where clashes continue with Daesh," said Mohamad Ghassri, spokesman for the forces of the UN-backed Government of National Accord (GNA), using another name for IS.
"The operation will not last much longer. I think we'll be able to announce the liberation of Sirte in two or three days," he told AFP.
Brett McGurk, US President Barack Obama's special envoy to the international coalition fighting IS, confirmed the advance.
"GNA aligned forces now making rapid advances against ISIL (IS) in Libya and beginning to enter its stronghold in Sirte," he tweeted.
Tightening the noose on IS, air strikes targeted the area around a conference centre where the jihadists had set up a command post, while the GNA's navy said it was in control of the waters off the eastern city.
A US Defense Department spokesman welcomed the advance.
"We certainly are encouraged by the progress we see those government forces making," Pentagon press secretary Peter Cook said.
"The fact that they are now under pressure in Libya... we think is a good thing and suggests that the Government of National Accord and the forces supporting that government are making progress."
Aziz Issa, a hospital spokesman in Misrata, east of Tripoli, said 115 fighters had been killed and 300 wounded in the anti-IS assault since mid-May.
Stepping up the operation, the GNA's navy has taken control of the coast of Sirte as part of the offensive, said Rida Issa, its commander for central Libya.
Signup to our newsletter and follow us on Facebook and Twitter!
"Our forces control the entire coast of Sirte. They (IS jihadists) will not be able to flee by sea," he told AFP.
Naval forces had supported the offensive, he said, including by "carrying out operations to open the way for ground forces to advance along the coast".
On Wednesday, the unity government said its forces had captured two barracks from the jihadists near Sirte, which IS has held since 2014.
'Turn against each other'
Analysts have advised caution over the decline of the Islamic State group.
"Soon IS will be driven out of Sirte. However, that definitely would not be the end of the group in Libya," said Mohamed Eljareh of the Rafik Hariri Center for the Middle East.
"If ISIS (IS) is defeated in Sirte, we expect an increase in attacks against oil installations to the south and also in the cities of Misrata and Tripoli."
Since launching the offensive against IS in mid-May, GNA forces have taken the town of Abu Grein, the power plant of Sirte and three barracks 20 kilometres from the city centre.
And on Thursday they announced the capture of the town of Harawa, 70 kilometres to the east of Sirte.
There has been no independent confirmation of the announced GNA advances over the past three weeks because of the absence of journalists on the ground.
Foreign intelligence services estimate IS has 5,000 fighters in Libya but its strength inside Sirte and the number of civilians living in the city are unavailable.
GNA forces are mostly made up of militias from western cities that have sided with the unity government of prime minister-designate Fayez al-Sarraj and the guards of oil installations that IS has repeatedly tried to seize.
Eljareh said the GNA still faced serious challenges even if it captured Sirte.
"What will happen to all the forces mobilised against IS," he asked. "There is a risk that they turn against each other."