ROME—The leaders of Italy’s hard-right League party and antiestablishment 5 Star Movement appeared close to forming a new government, possibly clearing the way for a euroskeptic alliance to come to power in the eurozone’s third-largest economy.

If it comes to fruition, a governing coalition between 5 Star, which is one of Europe’s biggest antiestablishment parties, and the League would hand antiestablishment groups in Europe one of their biggest victories to date.

The...