You can do this using the technique of geometric inversion. The map T defined by (inversion in the unit circle) takes circles through the origin to lines, and vice versa. It maps the Hawaiian earring to the family of vertical lines , and it maps the other family of circles to the family of vertical lines .

Let S be the map defined by , which takes the first family of lines to the second family. Then the composite map TST will do what you want.