Let $\{\varphi_k\}_{k=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle (OPUC) with respect to a probability measure $ μ$. We study the variance of the number of zeros of random linear combinations of the form $$ P_n(z)=\sum_{k=0}^{n}η_k\varphi_k(z), $$ where $\{η_k\}_{k=0}^n $ are complex-valued random variables. Under the assumption that the distribution for each $η_k$ satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when $\{\varphi_k\}$ are regular in the sense of Ullman-Stahl-Totik or are such that the measure of orthogonality $μ$ satisfies $dμ(θ)=w(θ)dθ$ where $w(θ)=v(θ)\prod_{j=1}^J|θ- θ_j|^{α_j}$, with $v(θ)\geq c>0$, $θ,θ_j\in [0,2π)$, and $α_j>0$, we give a quantitative estimate on the the variance of the number of zeros of $P_n$ in sectors that intersect the unit circle. When $\{\varphi_k\}$ are real-valued on the real-line from the Nevai class and $\{η_k\}$ are i.i.d.~complex-valued standard Gaussian, we prove a formula for the limiting value of variance of the number of zeros of $P_n$ in annuli that do not contain the unit circle.