I'd go for SQL (QGIS natively supports the SpatiaLite 4.3.0 dialect for any layer query) and check for the spatial relation with the negative buffered polygons:
The above query will return a table with all polygon
<id>s (paste the actual id field name here), their
geometries and a concatenated string of line
<id>s (same here) that intersect, but not touch, the negative buffered polygons of your layer.
Since you asked for meters, I put -1 in
ST_Buffer. However, the function will treat the passed number in CRS units; here, this implies that you are working with a CRS whose units are in meter! If you are working in LatLon, e.g. EPSG:4326, the parameter in
ST_Buffer will be treated as degrees...and thus makes no sense. In that case, a transformation into an appropriate projection is necessary; you can do that with
... ST_Buffer(ST_Transform(a.geometry, <srid_of_metric_CRS>), -1) ...