The people, who were not authorized to speak publicly, said she has started having conversations about a potential campaign.
Plans for Lindsey Graham's funeral were also announced on Friday. There will be a service in Washington on July 28 and more in South Carolina on July 29.
Darline Graham's eventual decision could dramatically shake up the scramble to fill her late brother's seat after he died last weekend. The filing period for a special primary runs from July 21 to July 28, and the primary is scheduled for Aug. 11.
Several other noteworthy politicians — including Reps. Russell Fry, Nancy Mace and Ralph Norman, as well as Lt. Gov. Pamela Evette — have been eyeing a run.
Darline Graham's conversations were first reported by Semafor.
South Carolina Gov. Henry McMaster appointed Darline Graham to serve the remainder of her brother's term, which ends in January. In his announcement on Monday, McMaster made no reference to her as a placeholder or symbolic appointment.
