“I’ve been thinking a lot about how we can move forward and how we can best honor Gabby’s strength, conviction and leadership,” Heinz said, according to the newspaper. “One of the best ways we can do that is by moving quickly and making sure that, in this very short time frame, we get somebody in there to fill that vacancy that is going to carry forward her tradition of common sense, moderate, consensus-building and problem solving.”

Giffords’s husband, Mark Kelly, has said she wants a “moderate” to fill her seat, but she has not said whether she plans on endorsing. Giffords resigned this month in order to focus on her ongoing recovery from an assassination attempt on her last January.

Heinz is an emergency-room physician and also served four years in the Arizona state legislature.

Last week, Republican Arizona state Sen. Frank Antenori announced he was running for the seat. Tea Party-backed Republican Jesse Kelly, who ran against Giffords in 2010, filed a statement of candidacy last Tuesday but has not formally announced his candidacy.

On Monday, Andrei Cherny resigned as the head of the Arizona Democratic Party and is expected to announce his run for the newly drawn Arizona ninth district. That decision would eliminate Cherny’s name from speculation that he might run for Giffords’s 8th district seat.

— Josh Lederman contributed to this report.