In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over
${\mathbb Z}$
is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set
$\Lambda $
and a fully Euclidean model set
with the property that finitely many translates of
cover
$\Lambda $
, we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in
$\Lambda $
if and only if k is at most the rank of the
${\mathbb Z}$
-module generated by
. We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets.