The sudden death of Senator Lindsey Graham, Republican of South Carolina, comes less than four months before he was set to appear on the ballot as a strong favorite to win a fifth term.

Now, South Carolina Republicans are scrambling to replace him on the general election ballot. The state leans heavily conservative and a new nominee would have a strong chance of becoming the state’s next senator.

Here’s how it will work, and who is in the mix:

Selecting a new nominee for the November ballot

The timeline set out in South Carolina law suggests that a special primary election will be held on Aug. 11.

Candidates can file for the seat as soon as July 21, the second Tuesday after Mr. Graham’s death, and the filing period will close on July 28. State law says that the election must happen on the second Tuesday after that deadline.

If no candidate secures a majority, a runoff between the top two finishers would take place two weeks later, on Aug. 25. (Mr. Graham, 71, who died of “a brief and sudden illness” Saturday evening, according to his office, had won the Republican primary by a wide margin earlier this year.)

Appointing a replacement in the Senate

Meanwhile, Gov. Henry McMaster, a Republican, can appoint a replacement to serve out the rest of Mr. Graham’s Senate term, through Jan. 3, 2027, under state law. The replacement could choose to compete for the Republican nomination in the special primary, or be a “caretaker” pick with no intention to stay in the seat in the long run.

An appointment decision could have implications for the special primary election, too. Mr. McMaster’s stamp of approval, for example, could help smooth the path for a special primary election contender.

Asked on the NBC show “Meet the Press” if there was someone he thought Mr. McMaster should appoint, President Trump said, “I have somebody that I think would be great.” He said he did not want to publicly disclose his preference, “because, you know, it’s too soon with Lindsey — I don’t want to even talk about anybody.”

There could also be implications beyond South Carolina. Republicans hold a narrow majority in the House, meaning that appointing a member of the lower chamber of Congress could pose risks for the party.

Representative Joe Wilson of South Carolina nodded to the slim edge in a social media post that appeared designed to remove himself from the conversation over who might replace Mr. Graham. In the post, he wrote that he had told Mr. Trump that his goal was “to remain in the House to keep his two-vote majority for the American people!!!”

A spokeswoman for Mr. McMaster said his office would address questions about filling the vacancy for Mr. Graham’s seat “when there are updates to share.”

“At this time, our focus is on honoring Senator Graham’s life and service,” the spokeswoman, Michelle LeClair, said.

A growing list of Republican names

The names of some elected officials are being mentioned in Republican circles as possible contenders for the special primary election. They include Representative Russell Fry, 41, an ally of Mr. Trump who holds a safely red seat in Congress, and Representative Ralph Norman, 73, a longtime lawmaker who is aligned with the hard-line conservative wing of the party.

Some jockeying already appeared to be underway on Sunday. Representative Nancy Mace is looking at a potential run for the seat, according to four people familiar with her thinking, who spoke on the condition of anonymity to describe private talks.

Ms. Mace finished a distant fifth in the Republican primary. She hinted at her ambition on social media on Sunday, posting a clip from the movie “The Godfather III” in which the character Michael Corleone says, “Just when I thought I was out, they pull me back in.”

Lt. Gov. Pamela Evette, a loyal ally of Mr. McMaster, is seen by some Republicans as a possible appointee. Mr. Trump initially backed her in last month’s Republican primary for governor, before later issuing a dual endorsement in the runoff between Ms. Evette and Attorney General Alan Wilson, who is Joe Wilson’s son. Alan Wilson went on to win.

Ms. Evette is also a possibility for the special election primary. A person close to Ms. Evette, speaking on condition of anonymity to disclose private conversations, said that the lieutenant governor had been receiving encouragement to consider running for the Senate seat.

“This could be an extraordinarily crowded, lifetime opportunity for a lot of up-and-comers,” said Walter Whetsell, a South Carolina Republican consultant.

An underdog Democrat weighs in

The Democratic nominee for Senate is Dr. Annie Andrews, a pediatrician who has been a strong fund-raiser in the race.

“I hope that South Carolinians will join me in setting partisanship aside and offering gratitude to Senator Lindsey Graham for his service to the great state of South Carolina,” Dr. Andrews said Sunday morning, praising him as a “man of great faith who proudly served our nation as a JAG officer and Air Force colonel.”

Michael Kruse and Emily Cochrane contributed reporting.