  In this paper we prove that every Stein manifold $S$ admits a proper
holomorphic immersion into any Stein manifold $X$ of dimension $2\mathrm{dim}S$
enjoying the density property or the volume density property. The case
$\mathrm{dim}S=1$ was proved beforehand by Andrist and Wold (Ann. Inst. Fourier
(Grenoble), 64(2):681-697, 2014). This result generalizes the classical theorem
of Bishop and Narasimhan for immersions to $\mathbb{C}^n$ with
$n=2\mathrm{dim}S$, and it complements the proper embedding theorem proved by
Andrist et al. when $\mathrm{dim}X>2\mathrm{dim}S$ (J. Anal. Math.,
130:135-150, 2016).
