Joining attribute of smaller polygon to attribute of biggest polygon when they are overlapping

I have two shapefile layers which include polygons. Each polygon on the first shapefile overlaps a polygon on the other shapefile.

I am trying to join the attribute of smaller polygons to the overlapping bigger polygons. I have used spatial join, but the attribute table of the bigger polygon is not showing the correct attributes of the smaller polygons inside it. I figured out there are some tiny polygons inside each polygon.

How can I join the attribute of the biggest overlapping polygon to the other shapefile?