The Turkish Army announced the city center was "completely surrounded," Turkey's Anadolu Agency reported on Tuesday.
Kurdish Afrin, in northwestern Syria, has been under the control of the People's Protection Units (YPG).
It comprises one of three cantons in Rojava, or Syrian Kurdistan.
Turkey considers the Kurdish groups in Afrin and Rojava as branches of the PKK, a named terror organization, and has framed its Operation Olive Branch as a counter-terror one.
More than 200 civilians have been reported killed in Afrin since Turkey launched its operation on January 20.
The United Kingdom has stated it believes Turkey's promises to protect civilians and "we will hold them to their statement."
Turkey has continued its operation despite a UN ceasefire resolution that agreed for a cessation of hostilities by all parties in Syria not fighting against ISIS or al-Qaeda.
More than 16,000 IDPs from Afrin are living in "tragic humanitarian conditions" in Aleppo's Nubl and Zahra, the Syrian Observatory for Human Rights reported on Monday.
After Monday's operations, Turkish forces and their partnered forces were only 4 kilometers away from the southwest of Afrin city, according to the Observatory. Additionally, the Turkish offensive reached "the outskirts" of Afrin city from the northern and eastern axes of advance.
The YPG, as part of the Syrian Democratic Forces (SDF), have formed the backbone of the US-led coalition's ground forces in the fight against ISIS.
Turkey's Foreign Minister Mevlut Cavusoglu announced that in his meeting with US State Secretary Rex Tillerson on Monday, they will ensure that YPG forces leave Manabí and that Ankara and Washington jointly run the city.
“We have reached a common understanding with the Americans on the issue of the stabilization of Manbij and other cities east of the Euphrates. We have established working groups for that,” Mevlut Cavusoglu said in an interview with the German newspaper Die Zeit.
Turkey and its Free Syrian Army proxies began their assault against Afrin on January 20.
This is a developing story... Last updated at 11:50 p.m.
