Hanafi
The sources give no view for it here.
كتابة المكاتب وعتقه
هل للمكاتب أن يكاتب أو يعتق؟
Each school’s answer
In our wordsHanafi
Not stated in the sources
Maliki
Not stated in the sources
Shafi'i
Not stated in the sources
Hanbali
Not stated in the sources
The texts
١٦٩٨ - مَسْأَلَةٌ: وَوَلَدُ الْمُكَاتَبِ مِنْ أَمَتِهِ حُرٌّ، وَكَذَلِكَ لَوْ مَلَكَ ذَا رَحِمٍ مَحْرَمَةٍ مِنْهُ، وَلَهُ أَنْ يُكَاتِبَ أَوْ يُعْتِقَ لِلنُّصُوصِ الْوَارِدَةِ فِي كُلِّ مَا ذَكَرْنَا - وَلَمْ يَخُصَّ اللَّهُ تَعَالَى مُكَاتَبًا مِنْ غَيْرِهِ - وَبِاَللَّهِ تَعَالَى التَّوْفِيقُ. [مَسْأَلَة ضمان نجوم الْكِتَابَة]
From the schools’ own books
Maliki
The sources give no view for it here.
Shafi'i
The sources give no view for it here.
Hanbali
The sources give no view for it here.
What is «in our words» is our standard wording; the books’ texts stay as they are.