Mr. Beshear is a two-term Democratic governor from Kentucky, a solidly Republican state. When he first won in 2019, his victory was regarded as a fluke; Mr. Trump won the state by about 30 percentage points three years earlier. But last year, he easily won re-election. He has emphasized the “common good,” working with Republicans on legislation, including medical marijuana, and often talks about his Christian faith. It’s a message that could be particularly appealing for Democrats now, as they try to win over moderate white voters in the suburbs.

Mr. Cooper is the governor of what might at best be called a swing-ish state — Barack Obama won there in 2008. Picking Mr. Cooper might put it in play, or at the least, press the Trump campaign to invest money in a state it would otherwise not have to worry about. In 2020, when Mr. Trump won the state by 1.3 percentage points, Mr. Cooper was elected by 4 points. But he is term-limited, so he cannot run again. And if Mr. Cooper gets the nod, every time he leaves the state to campaign, Lt. Gov. Mark Robinson, a very conservative Republican who is running for governor, becomes the acting governor.

The Arizona senator rose to national prominence after his wife, Representative Gabby Giffords, survived an assassination attempt in 2011. Mr. Kelly, a veteran of the Navy and a former astronaut, began campaigning for stricter gun control. He won his Senate seat in 2020. He has carved out an image as a moderate in Arizona, building a coalition that relied on white women in the suburbs as well as young Latino voters who were crucial in delivering the state to Mr. Biden in 2020.

Mr. Pritzker has been the governor of Illinois since 2019. That is not a battleground state, but he is a billionaire, and could help finance the campaign. That’s no small matter as Mr. Trump has seen an avalanche of financial support since Mr. Biden’s poor debate appearance in June. Mr. Pritzker, who is also Jewish, has drawn Democratic attention with his attacks on Mr. Trump. “Do they really want a president who is a felon who faces jail time?” Mr. Pritzker said in June.

Mr. Shapiro was elected as governor in 2022 after serving as the state’s attorney general. Pennsylvania is a must-win state for Democrats, and Mr. Shapiro’s stock among Democrats rose after he swamped his Republican opponent in 2022, Doug Mastriano, winning 56 percent of the vote.

As demonstrations against Israel spread across campuses this fall, Mr. Shapiro — who would be the second Jewish running mate on a major-party ticket in history if he were picked — denounced the rise in antisemitism after the attack by Hamas on Israel on Oct 7. He also won praise for overseeing quick repairs after a messy bridge collapse on I-95 last year.