  We study the interpolation analogue of the Hermite-Pad\'e type I
approximation problem. We provide its determinant solution and we write down
the corresponding integrable discrete system as an admissible reduction of
Hirota's discrete Kadomtsev-Petviashvili equations. Apart from the
$\tau$-function form of the system we provide its variant, which in the
simplest case of dimension two reduces to the non-autonomous discrete-time Toda
equations.
