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We establish certain properties for the class $𝒰(\Omega ,{\zeta }_0)$ of universal functions in $\Omega$ with respect to the center ${\zeta }_0\in \Omega$, for certain types of connected non-simply connected domains $\Omega$. In the case where $ℂ/\Omega$ is discrete we prove that this class is $G_{\delta }$-dense in $H\left(\Omega \right)$, depends on the center ${\zeta }_0$ and that the analog of Kahane’s conjecture does not hold.