BERLIN—German Chancellor Angela Merkel’s ruling partners are reconsidering their support for a decadesold arrangement that puts Germany under the U.S. nuclear shield, a development that could further undermine the country’s already-tense relationship with the Trump administration.

The center-left Social Democrats, or SPD, have appointed a commission to re-evaluate their positions on strategic, foreign and security policy, including the merits of “nuclear sharing,” a Cold War-era agreement under which German warplanes would...