Uncertainty over whether President Biden will continue seeking re-election, despite his pledges to stay in the race, has sharpened the focus on Donald J. Trump’s political calculations for choosing his Republican running mate.

Some people close to the former president said privately they now want him to give more consideration to a young contender or to a person of color, a move that could counter the possibility that someone like Kamala Harris, the first woman and first woman of color to serve as vice president, could replace Mr. Biden at the top of the Democratic ticket.

Such a scenario could help at least a couple of Mr. Trump’s top contenders: Senator J.D. Vance of Ohio, who is the second-youngest member of the Senate, and Senator Marco Rubio of Florida, whose Cuban American heritage is central to his political biography.

But among those in Mr. Trump’s circle who believe that Mr. Biden’s disastrous debate performance makes a Republican victory more likely in November, there’s another train of thought: that Gov. Doug Burgum of North Dakota is now a stronger option. Mr. Burgum, also one of Mr. Trump’s top contenders, has executive experience from two terms in that office and a long career as a business executive.

Either way, the countdown to Mr. Trump’s announcement has begun. His campaign has already parked an airplane dedicated to the Republican vice-presidential nominee in an undisclosed hangar, awaiting Mr. Trump’s decision, according to one person familiar with the planning.

There is also increasing anticipation for Mr. Trump’s next two rallies.

One is planned on Tuesday at Mr. Trump’s property in Doral, Fla., in Mr. Rubio’s home county, Miami-Dade. Mr. Trump has been urging his team for nearly two years to hold a rally at the Doral property, where, as president, he pushed to host the Group of 7 summit until criticism from fellow Republicans convinced him to hold it elsewhere. Another rally is scheduled next Saturday, July 13, in Butler, Pa., not far from the border with Mr. Vance’s home state of Ohio.

Mr. Trump’s reluctance to name a running mate has, in recent days, been partly to avoid shifting the focus away from the president during a stretch when Mr. Biden is facing calls from within the liberal establishment and Democratic Party to step aside.

Scott Reed, a Republican strategist who played a central role in the running mate selection process for former Senator Bob Dole, the 1996 Republican presidential nominee, noted that Mr. Trump will have to choose soon, regardless of what Mr. Biden does. Mr. Trump’s pick will be formally nominated during the Republican National Convention, which starts July 15.

With that uncertainty, Mr. Reed said he would urge the former president to base the decision on who could best help him win Michigan, Wisconsin and Pennsylvania. This trio of battleground states is known as the “blue wall” because of the crucial role each plays in the electoral map for Democrats.

“At this point, Trump should double down and do everything he can to climb the blue wall,” Mr. Reed said. “Burgum and Vance can help him in the Midwest, and that’s probably the new prism he’s looking through for this decision. It’s exactly what I would do.”

For months, Mr. Trump has insisted that he has not heavily weighed the political upsides of his running mate contenders and vowed that whomever he chooses will be ready to take over in the White House from him if needed.

For Mr. Trump, more significant priorities have been candidates who can help raise money, demonstrate discipline on the campaign trail, are unlikely to steal his precious spotlight and would fare well in a vice-presidential debate.

Mr. Trump believes the election this year will be decided by the names at the top of the tickets, a theory — rooted more in pragmatism than ego — that Mr. Biden and Mr. Trump are so well-known among voters that a running mate is unlikely to change those perceptions.

While that view is inherently tied to Mr. Biden remaining in the race, a different opponent may not have much of an effect on Mr. Trump’s vice-presidential calculations. Polls have signaled that preventing the former president from winning a second term is a bigger priority for many Democratic voters than making sure Mr. Biden is re-elected.