Mr. Biden faces a situation unique in the annals of modern American politics. He is the vice president, the highest-ranking member of his party interested in running for president, yet he is not the heir apparent. While every sitting vice president who sought it in the last half-century captured his party’s nomination, Mr. Biden would start as the underdog if he ran against Mrs. Clinton, the former secretary of state. …

Aides said Mr. Biden would do those things regardless of his own ambitions. And the preliminary moves do not address the more fundamental challenge: How would he run against Mrs. Clinton, who seemed to be Mr. Obama’s anointed heir in a joint interview they gave to the CBS program “60 Minutes” earlier this year? Mr. Biden would not have exclusive claim to the Obama legacy and, like any vice president, he would need to distinguish himself from the departing president. He could run to the left of her, emphasizing a more populist approach. Or he could run as the insurance policy in case she does not run or if she falters.

It is not clear that a presidential campaign makes sense for Mr. Biden, who would turn 74 shortly after the 2016 general election and 82 before the end of a second term.

But it is in his interest to keep the option open as long as possible to maintain his influence and to see what Mrs. Clinton decides to do. The moment Mr. Biden declares he will not run, he becomes a lame-duck No. 2 in a lame-duck administration.