They have been fighting to take the city, about 30km (20 miles) south of Turkey, from IS since late 2016.
Ten Turkish soldiers have been killed since Wednesday in the fighting.
It is part of a broader offensive by Turkey to push IS and Kurdish fighters away from Turkey's southern border.
Syrian forces meanwhile have closed in on al-Bab from the south, where there is reported to be fierce fighting.
Syrian troops and the pro-Turkish rebels supported by Turkish soldiers are now within 3km of each other on opposite sides of al-Bab, the Associated Press news agency says.
Although Turkey opposes Syrian President Bashar al-Assad in the war, reports suggest the two sides have reached an accommodation over al-Bab, orchestrated by Syria's key ally, Russia.
Syria is expected to be high on the agenda when the new director of the US Central Intelligence Agency (CIA), Mike Pompeo, holds with Turkish officials in Ankara on Thursday.
The visit is Mr Pompeo's first trip abroad since taking office. While the US and Turkey are part of the same anti-IS coalition, Turkey objects to US support for the predominantly Kurdish Syria Democratic Forces (SDF), whose People's Protection Units (YPG) contingent is regarded as a terrorist group by Turkey.
Turkish and rebel sources, and a UK-based monitoring group, said their forces had taken areas in the western outskirts of al-Bab on Wednesday.
A rebel commander told Reuters news agency that fighters from a pro-Turkish division of the Free Syrian Army were moving forward on Thursday from territory they had taken the day before.
The advances come after Syrian forces cut the main IS supply route into al-Bab, the group's last major stronghold in Aleppo province, earlier this week.
Five Turkish soldiers were killed on Thursday, and five the previous day, Turkish media say.
Turkish-backed Syrian rebels have captured the western outskirts of the Syrian city of al-Bab, held by so-called Islamic State (IS), reports say.