Following the suppliers' request for a judicial review of the act, last month the High Court rejected four out of five of their claims.
The suppliers had argued the act was "rushed through" and had "insufficient scrutiny", and they raised concerns that measures to tackle net piracy could be in breach of "basic rights and freedoms".
Last week, BT and TalkTalk said they were seeking leave to appeal the review. They intend for these "remaining areas of concern" to be looked at by the Court of Appeal.
BT and TalkTalk said they “felt obliged to seek clarity”, partly in case any third party questions the Act in future and has it overturned.