WASHINGTON—President Barack Obama will make the case for a new phase in nuclear-weapons reductions that would reduce arsenals by another one-third in a major foreign-policy speech in Berlin on Wednesday, a senior Obama administration official said.

Despite mounting tensions with Cold War adversary Russia over the Syrian civil war and missile defense, Mr. Obama is expected in the speech to say the U.S. and Russia can significantly reduce their arsenals beyond the cuts mandated under the 2010 New Start, or Strategic Arms Reduction...