Mr. Bryant has not offered any hints about whom he would appoint to replace Mr. Cochran, but Republicans close to the governor said he would not choose Mr. McDaniel.

Immediately after Mr. Cochran put out word on Monday that he would leave the Senate after nearly four decades, the senator’s top aide and close adviser, Brad White, said, “I hope he appoints somebody who can win a general election.”

Asked if that includes Mr. McDaniel, Mr. White said, “No.”

President Trump and Senator Mitch McConnell, Republican of Kentucky and the majority leader, have encouraged Mr. Bryant to appoint himself, arguing that the party needs to block Mr. McDaniel and forestall any potential replay of the Roy S. Moore debacle last year in Alabama that handed the Democrats a seat in the Deep South.

Mr. Bryant has to date resisted the entreaties. If he does not take the seat, the governor is expected to consider Lt. Gov. Tate Reeves, who is also thought to want to run for governor when Mr. Bryant becomes term-limited in 2019, and Secretary of State Delbert Hoseman. Other possibilities include the State House speaker, Philip Gun, and the state auditor, Stacey Pickering.

Democrats have not won a Senate race in Mississippi since 1982, when the veteran Senator John Stennis defeated Haley Barbour, but state and national Republicans worry about whether their grip on the state’s two seats could be in peril should Mr. McDaniel find success. Last week, David Baria, the State House Democratic leader, announced his intention to run for Mr. Wicker’s seat. Democrats have also wooed Brandon Presley, a cousin to Elvis Presley and the chairman of the state’s Public Service Commission, to run for the Senate.

Mike Espy, a former Democratic congressman and agriculture secretary in the Clinton administration, said late Monday that he would consider running for Mr. Cochran’s seat.

Mr. Espy, who is African-American and from a prominent Delta family, could be a formidable candidate in a state where black voters make up much of the Democratic Party.