Roll Call reported Monday two sources said conservatives have approached Mark Sanford about running and he is considering it. He served in the House in the 1990s before his terms as governor.
Scott is giving up his seat in the heavily Republican 1st District after being appointed by Gov. Nikki Haley to take Sen. Jim DeMint's seat. DeMint resigned this month to run the conservative Heritage Foundation think tank in Washington.
Sanford served as governor from 2003 until his resignation in 2011 amid a scandal about an extramarital affair.
Roll Call and Politico both said Jenny Sanford, who ran her husband's campaigns for the 1st District House seat, also is being mentioned as a possible candidate.
Politico said other potential contenders include state House Majority Whip Jimmy Merrill, state Rep. Chip Limehouse, state Sens. Larry Grooms and Chip Campsen, and Carroll Campbell III, son of former Gov. Carroll Campbell Jr.
"It's going to be a free-for-all," Republican consultant Chip Felkel told Politico.
Under South Carolina law, a special primary election is to be held 11 weeks after a resignation, with a runoff election if no candidate gets more than 50 percent of the vote. A general election is to be held 18 weeks after the vacancy occurs.