  The large-N limit of the two-dimensional non-local U$(N)$ Yang-Mills theory
on an orientable and non-orientable surface with boundaries is studied. For the
case which the holonomies of the gauge group on the boundaries are near the
identity, $U\simeq I$, it is shown that the phase structure of these theories
is the same as that obtain for these theories on orientable and non-orientable
surface without boundaries, with same genus but with a modified area
$V+\hat{A}$.
