  We evaluate the valence double parton distribution (dPDF) of the pion in the
Nambu--Jona-Lasinio model. At the low-energy quark-model scale and in the
chiral limit a particularly simple factorized form $D(x_1,x_2, \vec{q}) =
\delta(1-x_1-x_2) F(\vec{q})$ follows, where $x_{1,2}$ denote the longitudinal
momentum fractions of the valence quark and antiquark, and $\vec{q}$ is their
relative transverse momentum. For $\vec{q}=\vec{0}$ our result complies to the
Gaunt-Sterling sum rules. We carry out the necessary dDGLAP evolution to higher
scales via the Mellin moments and explore its impact on the correlation defined
as the ratio of dPDF to the product of single parton distributions, $D(x_1,x_2,
\vec{q}=\vec{0})/D(x_1)D(x_2)$. Since the ratios of the valence Mellin moments
$\langle x_1^n x_2^m \rangle / \langle x_1^n \rangle \langle x_2^m \rangle $
are invariants of the dDGLAP evolution, they may serve as robust measures of
these correlations. Model predictions, which can be tested in the upcoming
lattice simulations, are provided. We also discuss the transverse form factor
related to the dPDF of the pion.
