Let m be a positive integer. We show that for any given real number ${alpha};{in};[0,;1]$ and complex number $mu$ with $|mu|{leq}1$ which satisfy $e^{2{pi}i{alpha}}{mu}^m;{ eq};1$, there exists a Blaschke product B of degree 2m + 1 which has a fixed point of multiplier ${mu}^m$ at the point at infinity such that the restriction of the Blaschke product B on the unit circle is a critical circle map with rotation number $alpha$. Moreover if the given real number $alpha$ is irrational of bounded type, then a modified Blaschke product of B is quasiconformally conjugate to some rational function of degree m + 1 which has a fixed point of multiplier ${mu}^m$ at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point.