ArcGIS: Selecting features from a grid layer which are on the edge of another feature by location

I have two layers which come from the same data source. The first one is a vector grid layer which contains 1km x 1km features and looks like this:

The second one is the grid layer dissolved into Sub-Bains, which i.e. looks like this:

I need to select all the features from layer one which touch the borderline of each Sub-Basin of layer two from within the Sub-Basin. For example of this Sub-Basin:

This has to be done for multiple (1000) Sub-Basins, so selecting them manually would be too time consuming.
To include some context: I need to identify the feature with the biggest water outflow into another Sub-Basin and for that I need to export all the features which are on the inner edge of the basin.

Is there a tool or a way this can be done?