Could Mark Kelly run for wife's congressional seat?

A Democratic official told Fox News on Sunday that he’s confident the party will hold the seat being vacated by Rep. Gabrielle Giffords, who announced her resignation from Congress effective this week.

But out of deference to the congresswoman, the official and another Democratic official in Washington each refused to speculate on names that would be put up for either the special election to be held in the spring or the regular November general election. Each said that process would move quickly in the days ahead.

One Democratic name that keeps popping up, however, is Giffords' husband, astronaut Mark Kelly. Kelly has said on more than one occasion that he's not interested in the seat, and is focused on Giffords' recovery.

Kelly has been a vocal advocate of Giffords' continuing her public service after she recovers from a gunshot wound that injured her more than a year ago while she was conducting a town hall in Tucson. Six people, including Giffords' aide, were killed and 13 were injured.

Political folks in Arizona have been waiting for an announcement on what Giffords would do. Democrats have held back from pushing forth challengers in the open Senate seat being vacated by Jon Kyl while awaiting Giffords' decision. Republicans, with one exception -- state Sen. Frank Antenori, have held off from announcing challenges for the House seat held by Giffords.

Antenori has had an exploratory committee for several months. A person close to Antenori told Fox News that a decision on whether to move forward could come in the next week, but Antenori also doesn’t want to take away from Giffords or her work this week.

The Democratic official in Arizona told Fox News that the special election to replace Giffords will be based on the current boundaries of Arizona's 8th District while the November election will be held under the new redistricting map.

Fox News' Chad Pergram contributed to this report.