Giffords, an Arizona Democrat, is still recovering eight months after being shot in the head in Tucson in January.
She has not announced whether she will seek re-election.
Republican State Senator Frank Antenori is forming an exploratory committee to look into a possible run for the seat.
Antenori says he will make his decision early next year.
Giffords is in her third term in Congress. She has been mentioned as a potential candidate for the U.S. Senate
The Associated Press contributed to this report.