According to Rabbi Prof. Yoel Elitzur in his essay on this question, the Bavli's understanding seems to come from the list of cities in Divrei Hayamim, where it says, for example (1:6:42):

"וְלִבְנֵי אַהֲרֹן נָתְנוּ אֶת עָרֵי הַמִּקְלָט אֶת חֶבְרוֹן וְאֶת לִבְנָה וְאֶת מִגְרָשֶׁיהָ וְאֶת יַתִּר וְאֶת אֶשְׁתְּמֹעַ וְאֶת מִגְרָשֶׁיהָ"

While in Yehoshua it says (Yehoshua 21:13):

"וְלִבְנֵי אַהֲרֹן הַכֹּהֵן נָתְנוּ אֶת עִיר מִקְלַט הָרֹצֵחַ אֶת חֶבְרוֹן וְאֶת מִגְרָשֶׁהָ וְאֶת לִבְנָה וְאֶת מִגְרָשֶׁהָ"

The key difference being that in Yehoshua only the chosen six cities are referred to as a city of refuge, while the phrase "cities of refuge" in Divrei Hayamim seems to refer also to the rest of the cities, and not only the original six.

However, he also explains that this view was not widely-accepted, neither in the Bavli - the previous sugiya hinges on the understanding that there are only six cities - nor in the Yerushalmi (Makkot 7b), where they go to great lengths to explain that in the future there will be at most 15 cities of refuge.

According to Rabbi Prof. Yoel Elitzur in his essay on this question, the Bavli's understanding seems to come from the list of cities in Divrei Hayamim, where it says, for example (1:6:42):

"וְלִבְנֵי אַהֲרֹן נָתְנוּ אֶת עָרֵי הַמִּקְלָט אֶת חֶבְרוֹן וְאֶת לִבְנָה וְאֶת מִגְרָשֶׁיהָ וְאֶת יַתִּר וְאֶת אֶשְׁתְּמֹעַ וְאֶת מִגְרָשֶׁיהָ"

While in Yehoshua it says (Yehoshua 21:13):

"וְלִבְנֵי אַהֲרֹן הַכֹּהֵן נָתְנוּ אֶת עִיר מִקְלַט הָרֹצֵחַ אֶת חֶבְרוֹן וְאֶת מִגְרָשֶׁהָ וְאֶת לִבְנָה וְאֶת מִגְרָשֶׁהָ"

The key difference being that in Yehoshua only the chosen six cities are referred to as a "city of refuge", while the phrase "cities of refuge" in Divrei Hayamim seems to refer also to the rest of the cities, and not only the original six.

However, he also explains that this view was not widely-accepted, neither in the Bavli - the previous sugiya hinges on the understanding that there are only six cities - nor in the Yerushalmi (Makkot 7b), where they go to great lengths to explain that in the future there will be at most 15 cities of refuge.