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Others on Haley's list include Reps. Tim ScottTimothy (Tim) Eugene ScottHow expanded credit data can help tackle inequities Dems erupt over GOP 'McCarthyism' as senators vet Biden bank watchdog pick Why Democrats' prescription drug pricing provision would have hurt seniors MORE (R-S.C.) and Trey GowdyTrey GowdyTrey Gowdy sets goal of avoiding ideological echo chamber with Fox News show Fox News signs Trey Gowdy, Dan Bongino for new shows Pompeo rebukes Biden's new foreign policy MORE (R-S.C.) as well as former South Carolina Attorney General Henry McMaster (R). Scott's name has been the one to receive the most attention.

Haley continued to play coy about a decision on Thursday, posting the lyrics to Adele's "Rumour has it" on her Facebook page without mentioning the seat.

Sources say she will pick a replacement for DeMint soon, though she has given no timeline for a choice. Whoever is chosen will be up for election in 2014, and could face a primary challenge.