For a finite subset ${Lambda}$ of $mathbb{N}_0{ imes}mathbb{N}_0$, where $mathbb{N}_0$ is the set of nonnegative integers, we say that a complex data ${gamma}_{Lambda}:={{gamma}_{ij}}_{(ij){in}{Lambda}}$ in the unit disc $mathbf{D}$ of complex numbers has a cyclic subnormal completion if there exists a Hilbert space $mathcal{H}$ and a cyclic subnormal operator S on $mathcal{H}$ with a unit cyclic vector $x_0{in}mathcal{H}$ such that ${langle}S^{*i}S^jx_0,x_0{ angle}={gamma}_{ij}$ for all $i,j{in}mathbb{N}_0$. In this note, we obtain some sufficient conditions for a cyclic subnormal completion of ${gamma}_{Lambda}$, where ${Lambda}$ is a finite subset of $mathbb{N}_0{ imes}mathbb{N}_0$.