In this paper, we present a linear-time approximation scheme for $k$-means clustering of \emph{incomplete} data points in $d$-dimensional Euclidean space. An \emph{incomplete} data point with $Δ>0$ unspecified entries is represented as an axis-parallel affine subspaces of dimension $Δ$. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for $k$-means clustering of axis-parallel affine subspaces of dimension $Δ$ that yields an $(1+ε)$-approximate solution in $O(nd)$ time. The constants hidden behind $O(\cdot)$ depend only on $Δ, ε$ and $k$. This improves the $O(n^2 d)$-time algorithm by Eiben et al.[SODA'21] by a factor of $n$.