  Enumerating matchings is a classical problem in the field of enumeration
algorithms. There are polynomial-delay enumeration algorithms for several
settings, such as enumerating perfect matchings, maximal matchings, and
(weighted) matchings in specific orders. In this paper, we present
polynomial-delay enumeration algorithms for maximal matchings with cardinality
at least given threshold $t$. Our algorithm enumerates all such matchings in
$O(nm)$ delay with exponential space, where $n$ and $m$ are the number of
vertices and edges of an input graph, respectively. We also present a
polynomial-delay and polynomial-space enumeration algorithm for this problem.
As a variant of this algorithm, we give an algorithm that enumerates all
maximal matchings in non-decreasing order of its cardinality and runs in
$O(nm)$ delay.
