Only Florida’s secretary of state can call for a recount, per state law.
DeSantis and his campaign have not yet commented on a potential recount in the governor's race. He and running mate Jeanette Nunez are scheduled for events in South Florida on Thursday afternoon.
DeSantis was supported in the race by President Donald Trump.
Gillum conceded Tuesday night before a crowd gathered on the campus of Florida A&M University. In his concession speech, Gillum told supporters, "This was a difficult task, and I am sorry I couldn't bring it home for you."
The Tallahassee mayor had sought to become the state's first black governor and the first Democrat to win the governor's race in more than 20 years.
The 39-year-old pulled off an upset when he won the Democratic primary in August.
Florida's race for U.S. Senate seems to be headed toward a recount. Republican Rick Scott leads longtime Democratic Sen. Bill Nelson by only 22,000 votes as of Thursday afternoon, which is well within the 0.5 percent required to start a state-mandated recount.