Rose's four-year term concluded last weekend.
Sanford asked Rose on Friday to serve as acting director until a replacement is appointed, Sanford's spokesman Will Folks said Monday.
"The governor had several conversations with Director Rose last week," Folks said. "We expect to make an announcement soon regarding a permanent replacement."
Folks would not say who is being considered for the post or explain why Sanford had not selected a replacement before Rose's term ended.
"The governor has always been very deliberate in the process of making appointments," Folks said.
Rose was unavailable to comment Monday, according to spokesman Sid Gaulden.
Sanford said during a gubernatorial debate in the 2002 campaign it was "a waste of human life" that so many people were forced to stand in line for hours at Division of Motor Vehicle offices, then under Rose's control.
The division has since been made a separate state agency, under Sanford's control, and changed its name to the Department of Motor Vehicles.
DMV was getting extensive complaints because of "Project Phoenix," a $40 million computer changeover that led to long lines and slow service initially.