Russia has led a fresh push for diplomacy to resolve the Syrian conflict in recent months and Syrian opposition leaders say they have detected a shift in Moscow’s long-standing support for President Bashar al-Assad.

Russian officials are showing more openness to discussing alternatives to Mr. Assad as his regime loses territory, the opposition leaders say. And with Russia’s economy battered by a plunge in oil prices and Western sanctions, the government may be considering both the strategic and economic benefits of changing...