"Turkish forces and allied rebels in the Euphrates Shield campaign entered the western edge of the town and took control of a number of areas," the Britain-based Syrian Observatory for Human Rights said.
The monitor said heavy clashes were underway with IS in the town, which is the jihadist group's last stronghold in the northern province of Aleppo.
The town is the target of two competing assaults, with the Turkish-led "Euphrates Shield" campaign advancing from the north, east and west, while Syrian government troops attack from the south.
It has been besieged since Monday, when Syrian troops severed a road leading into the town from the south.
By Friday, government forces were just 1.5 kilometres (less then a mile) from the southern outskirts of Al-Bab.
Turkey began an unprecedented campaign inside Syria in August, targeting both IS and Kurdish militia.
After initial rapid progress, the campaign has been mired since December in the deadly fight for Al-Bab.
66 Turks killed in campaign
Turkey's Dogan news agency says 66 Turkish soldiers have been killed in the campaign since it started, mostly in IS attacks.
And on Thursday, three Turkish soldiers were killed when a Russian air strike accidentally hit their position in a strike targeting IS fighters in Al-Bab.
Moscow said the incident was an accident and is under investigation.
Despite backing opposite sides in Syria's conflict -- Moscow is a government ally while Turkey supports the opposition -- the two countries have worked closely in recent months.
They helped broker a nationwide ceasefire in place since December 30, and sponsored a round of peace talks last month in the Kazakh capital, Astana.
Al-Bab has been under IS control since 2014, when the group seized large swathes of territory in Syria and neighbouring Iraq, proclaiming its self-described caliphate.
In recent months, the jihadists have been rolled back in large parts of northern Syria, both by the Turkish campaign but also a Kurdish-Arab alliance known as the Syrian Democratic Forces (SDF).
The SDF fights with air support from the US-led coalition battling IS in Syria and Iraq, but Turkey regards the Kurdish component of the SDF as "terrorists".
The alliance is pushing towards IS's de facto Syrian capital Raqa in an operation dubbed "Wrath of the Euphrates".
The advance has progressed slowly, in part, SDF officials say, because IS has heavily mined territory around Raqa.
New talks in Astana?
The Observatory said Saturday that SDF fighters had now advanced to around eight kilometres from the eastern outskirts of Raqa, though their forces are further from the north of the city.
Turkey has suggested that it could turn its sights to Raqa after the Al-Bab operation is complete, with Turkish President Recep Tayyip Erdogan discussing both Al-Bab and Raqa in a call with US President Donald Trump this week.
Syria's conflict has killed more than 310,000 people since it began with anti-government protests in March 2011.
Successive rounds of peace talks, including discussions organised by Russia and Turkey in Kazakhstan last month, have failed to advance a political solution to the conflict.
A new round of UN-sponsored talks is scheduled to take place in Geneva on February 20, but invitations have yet to be sent out.
On Saturday, Kazakhstan's foreign ministry said Syrian government officials and rebels were being invited to new talks next week in the capital Astana.
"It is planned to hold the latest high-level meeting within the Astana process on resolving the situation in Syria on February 15 and 16," the ministry said in a statement.
It added that UN peace envoy Staffan de Mistura and US observers would also be invited to the talks.
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