PARIS—As Greece careens toward a potential rupture with the eurozone, France is emerging as the only friendly voice in a currency bloc where others are taking an increasingly hard line.

Paris has struck a dovish tone over months of tortuous talks with Greece, but left the heavy-lifting of negotiations to the institutions representing Athens’s creditors. In recent days, however, France has taken a more active role, according to people familiar with the matter, seeing its vision of the eurozone at risk of a heavy blow if Greece...