No, this only holds when $\varphi \in D(T)$.
It's enough to assume that $T$ is closed and densely defined, so that $T^{**} =T$. Let $\psi \in D(T^*)$ be arbitrary. If the hypothesis holds then we have $$ |\langle \varphi, T^* \psi \rangle| = \lim_{n \to \infty} |\langle \varphi_n, T^* \psi \rangle| = \lim_{n \to \infty} |\langle T \varphi_n, \psi \rangle| \le k(\varphi) \|\psi\|$$ which is exactly the definition of $\varphi \in D(T^{**}) = D(T)$.