He has been more purposeful behind the scenes, fellow lawmakers and Republican strategists in Georgia said. But neither Mr. Collins nor others interested in the Senate appointment are likely to take their efforts public until Mr. Kemp outlines what he is looking for in a replacement for Mr. Isakson. The governor has until December to fill the seat, and a person familiar with his thinking said he intends to more clearly lay out a timeline for the decision after the threat of Hurricane Dorian passes the state.

With Democrats eager to try to flip at least one of the state’s Senate seats, Republicans and political strategists in the state believe Mr. Kemp will be looking to appoint someone who is ready to run two expensive races in quick succession and can help the party ticket statewide.

“Governor Kemp has a really tough decision to make,” said Heath Garrett, a former top aide to Mr. Isakson and a political strategist in the state. “Senator Isakson and his team are going to be very respectful of that decision. Not only is he picking somebody to run in 2020 but picking what is close to being a running mate in 2022.”

Mr. Collins is not the only House member whose name is circulating. Representative Tom Graves, who has represented northwest Georgia since 2010, is thought to be a serious contender, as is Karen Handel, a one-term congresswoman who narrowly lost a suburban Atlanta seat last fall. Mr. Graves has also spoken with Mr. Kemp and begun quietly gauging support around the state, according to a congressional aide familiar with Georgia politics.

Potential candidates among current and former statewide office holders include Lt. Gov. Geoff Duncan, state Attorney General Chris Carr and Sonny Perdue, a former Georgia governor who now serves as Mr. Trump’s agriculture secretary.

Close watchers of Georgia politics have also speculated that Mr. Kemp could aim to broaden the Republican coalition in the Atlanta suburbs, where Mr. Trump and Mr. Kemp underperformed compared with past Republicans, by appointing someone other than a white man. Possibilities include Harold D. Melton, the chief justice of the Supreme Court of Georgia, who is black; BJay Pak, the United States attorney for Northern Georgia and a former state representative who is of Korean descent; and Kelly Loeffler, a business executive who considered a Senate bid in 2013.

Mr. Collins has served in the House since 2013 and ascended to the top Republican position on the Judiciary Committee in January. The position has dramatically boosted his public profile, securing him a recurring slot on Fox News and the first word for his party in high-stakes hearings convened by Democrats on the findings of the special counsel, Robert S. Mueller III, as they try to build a case for impeachment.