In this paper, we study the flexibility of embeddings of circular graphs \(C(2n,2)\), \(n \geq 3\) on the projective plane. The numbers of (non-equivalent) embeddings of \(C(2n, 2)\) on the projective plane are obtained, and by describing structures of these embeddings, the numbers of (non-equivalent) weak embeddings and strong embeddings of \(C(2n, 2)\) on the projective plane are also obtained.