For an $N \times N$ random unitary matrix $U_N$, we consider the random field defined by counting the number of eigenvalues of $U_N$ in a mesoscopic arc of the unit circle, regularized at an $N$-dependent scale $ε_N>0$. We prove that the renormalized exponential of this field converges as $N \to \infty$ to a Gaussian multiplicative chaos measure in the whole subcritical phase. In addition, we show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in \cite{Ost16}. By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. The proofs are based on the asymptotic analysis of certain Toeplitz or Fredholm determinants using the Borodin-Okounkov formula or a Riemann-Hilbert problem for integrable operators. Our approach to the $L^{1}$-phase is based on a generalization of the construction in Berestycki \cite{Berestycki15} to random fields which are only \textit{asymptotically} Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context.