  The discrete Painlev\'e III equation is investigated based on the bilinear
formalism. It is shown that it admits the solutions expressed by the Casorati
determinant whose entries are given by the discrete Bessel function. Moreover,
based on the observation that these discrete Bessel functions are transformed
to the $q$-Bessel functions by a simple variable transformation, we present a
$q$-difference analogue of the Painlev\'e III equation.
