Four Republican candidates are competing to be the party's nominee -- Frank Antenori, Jesse Kelly, Martha McSally and Dave Sitton, the Arizona Daily Star in Tucson reported.
Democrat Ron Barber and Green Party candidate Charlie Manolakis are unopposed on their party ballots.
Giffords, a Democrat, officially resigned from Congress in January to focus on recovering from a gunshot wound she got during a 2011 shooting spree. Giffords was shot in the head during a meet-and-greet event in January 2011 outside a Tucson grocery store. Six people were killed and 13 people, including Giffords, were injured.
The special election is June 12.