This would be tantamount to declaring political war. Yet after more than seven years of wrestling with Republicans, Mr. Obama may think that he has little to lose by provoking them, especially if he is reasonably confident that a Democrat will win the White House in November. He could choose from among the younger stars in the party, like Kamala D. Harris, California’s attorney general. He might select a left-leaning judge like Diane P. Wood of the Court of Appeals for the Seventh Circuit in Chicago. Or he could opt for the dream candidate of the left: Senator Elizabeth Warren of Massachusetts.

Image Some Democrats believe choosing Senator Elizabeth Warren would unify liberals. Credit... Zach Gibson/The New York Times

Some Democrats believe that the nomination of Ms. Warren, a deeply admired voice in the party on economic inequality, could be a unifying move after what is turning out to be a bruising battle between Mrs. Clinton and Mr. Sanders. Such a nomination could be a balm for liberals and young people if Mr. Sanders loses the nomination, given that he and Ms. Warren are strong critics of corporate malfeasance and public corruption. While Ms. Warren is a Harvard Law School professor, she may be reluctant to put herself through a nasty confirmation battle, and it is not clear that Mr. Obama holds her in high regard: He did, after all, pass up the chance to nominate her to head the Consumer Financial Protection Bureau.

Still, the selection of such a strong progressive voice would energize Democratic and Republican voters because of the possibility of a five-member liberal majority on the Supreme Court.

Make It Awkward for Republicans

This may be the most far-fetched choice in an era of polarization, but Mr. Obama could put Republicans in an awkward position by nominating an admired Republican figure.

This may lure a few votes from Republicans while also suggesting that the president was acting above the partisan political fray. And it could appeal to those independent voters or moderate Republicans who are tired of Washington gridlock. Using a court pick to woo voters from the opposition is not without precedent: President Dwight D. Eisenhower, a Republican, used a recess appointment to nominate William J. Brennan Jr., a Roman Catholic Democrat from the Northeast, in October 1956. (The next month, Eisenhower carried New Jersey, Brennan’s home state, along with the rest of the Northeast.)

Image Picking Gov. Brian Sandoval of Nevada, a moderate Republican, could hem in his party. Credit... Cathleen Allison/Associated Press

In the remote possibility that he decides to go down this path, Mr. Obama would appear to have two options. He could name an older Republican who would not serve on the court for long. (Senator Orrin G. Hatch, a former chairman of the Judiciary Committee and the chamber’s longest-serving Republican, will turn 82 next month.) Or he could tap a younger Republican who is politically moderate enough to appeal to Democrats but would still hem in Republicans. Gov. Brian Sandoval of Nevada supports abortion rights and, after the court’s same-sex marriage decision last year, said his state’s arguments “against marriage equality are no longer defensible.” He is also from a fast-growing and increasingly diverse swing state, is Hispanic and was state attorney general and a federal judge before becoming governor.