A jpg does not accommodate a transparent area. Open the file. Since the image is on the Background area, trash the lock. Then place a layer below it (Cmd click on new layer symbol at bottom of Layers panel.). Return to the image layer: select and delete the chosen area. Select > Deselect the selection, trash the bottom layer and save as a Photoshop (psd file).
Thanks. I just realized as I read your first sentence that the image I'm wanting to alter was actually originally modified in Illustrator. So, I will repost my question in that forum so as not to mislead anyone here.