BEIRUT (Reuters) - The United States is trying to avert a greater war between Israel and Lebanon's Hezbollah movement, U.S. envoy Amos Hochstein said on Tuesday, following an escalation in cross-border fire between the foes along Lebanon's southern frontier.

Iran-backed Hezbollah has been trading fire with Israel for the last eight months in parallel with the Gaza war. Last week, the group fired the largest volleys of rockets and drones of the hostilities so far at Israeli military sites, after an Israeli strike killed the most senior commander yet.

Hochstein, special envoy to U.S. President Joe Biden, said he had been dispatched to Lebanon immediately following a brief trip to Israel because the situation was "serious".

"We have seen an escalation over the last few weeks. And what President Biden wants to do is avoid a further escalation to a greater war," Hochstein said on Tuesday.

He had met with the head of Lebanon's army earlier on Tuesday morning and spoke to reporters following a meeting with parliament speaker Nabih Berri, who heads the armed Amal movement, which is allied to Hezbollah and has also fired rockets on Israel in recent months.

The U.S. and France are engaged in diplomatic efforts to secure a negotiated end to the hostilities along Lebanon's border. Hezbollah says it will not halt its attacks unless there is a ceasefire in the Gaza Strip.

Hochstein on Tuesday urged Hamas to accept a U.S.-backed proposal for a ceasefire in Gaza, which he said "also provides an opportunity to end the conflict across the Blue Line", a reference to a demarcation line between Lebanon and Israel where parts of the international border are disputed.

Hochstein is also set to meet caretaker Lebanese Prime Minister Najib Mikati on Tuesday.

(Reporting by Maya Gebeily; Editing by Alex Richardson and Peter Graff)