Sen. Darline Graham of South Carolina, who was appointed to fill the vacancy left by the death of her brother, GOP Sen. Lindsey Graham, is engaged in serious discussions about running for his seat in the upcoming special Republican primary on Aug. 11.

On Thursday, the new senator indicated in a White House meeting that she’s weighing a bid, CBS News confirmed, according to two sources familiar with her plans. Semafor first reported that she’s considering a Senate run.

South Carolina Gov. Henry McMaster on Monday appointed her as Lindsey Graham’s replacement for the remainder of his current term, which ends in January. Darline Graham, 62, was sworn in Tuesday and is the first woman to represent South Carolina in the Senate. 

Her fellow South Carolina senator, Tim Scott, who also chairs the fundraising arm for Senate Republicans, the National Republican Senatorial Committee, seemed open to the idea when asked about it Wednesday by CBS News. 

“Darline has so far been off to a remarkable start,” Scott said. “Why not her?”

Lindsey Graham, whose sudden death last weekend shocked the Senate, had served in the upper chamber since 2003 and was widely expected to win his fifth term this November. 

Darline Graham, who has had a career in disability services and most recently served as commissioner of the South Carolina Commission for the Blind, doesn’t have a background in elected office. But she said Monday that with the support of her brother’s staff, she feels confident in her ability to carry out the role.