Haley said Monday she would not appoint a placeholder to succeed DeMint, who announced his resignation last week to lead the Heritage Foundation, to hold the seat until a 2014 special election, The (Columbia) State reported Tuesday.
U.S. Rep. Tim Scott, the presumed front-runner to succeed the conservative DeMint, said he hasn't spoken to the first-term Republican governor or her staff about the vacancy.
Scott wouldn't say if he'd accept an appointment if offered, but told The State he received numerous calls encouraging him to discuss it with Haley.
"Certainly, there is a lot of interest out there," he said.
If he were appointed, Scott would be the first black U.S. senator from the South since Reconstruction, The State said.
"He could be the new national face of South Carolina," GOP political consultant Richard Quinn of Columbia said.
In a statement Monday, Haley said she didn't want to "tie the hands" of an appointee by tapping someone to act as a placeholder until the special election.
"I do not want to deprive our state's citizens of the chance to render their judgment on the appointee's performance by way of their vote," Haley said. "Most importantly, while I am an avid supporter of term limits, I do not want the effectiveness of our state's new U.S. senator to be undermined by the fact that he or she will automatically be leaving the office such a very short time after assuming it."