GEOFF BENNETT: Today, the U.S.

said it's discussing a new Iranian proposal to reopen the Strait of Hormuz, but pause all negotiations over at Iran's nuclear program.

AMNA NAWAZ: And Lebanon marked the deadliest day since the fraying cease-fire in that country went into effect two weeks ago.

Israeli attacks killed more than a dozen people in the south.

Nick Schifrin reports on tensions running high on the Israeli-Lebanese border and what appears to be an impasse in the war in Iran.

NICK SCHIFRIN: In the Strait of Hormuz today, tankers at a standstill, as the war's diplomacy is in a standoff.

Today, Iran's top diplomat visited with Russian President Vladimir Putin, a longtime ally who called Iran's struggle heroic.

And Foreign Minister Abbas Araghchi blamed the U.S.

for a lack of progress.

ABBAS ARAGHCHI, Iranian Foreign Minister (through translator): The Americans' approaches caused the previous round of talks, despite the progress that had been made, not to reach its objectives, the excessive demands they made and the incorrect approaches they adopted.

NICK SCHIFRIN: But an Iranian official and a senior regional official tell PBS "News Hour" Iran has made the latest proposal to reopen the strait if the U.S.

lifts its blockade, unfreezes blocked Iranian assets and pauses any negotiation of Iran's nuclear program.

The Iranian official said the U.S.

has already described this offer as not good enough.

KAROLINE LEAVITT, White House Press Secretary: The proposal was being discussed.

NICK SCHIFRIN: Today, at the White House, spokesperson Karoline Leavitt did not reject the deal out of hand.

MARCO RUBIO, U.S.

Secretary of State: They're very good negotiators.

NICK SCHIFRIN: But Secretary of State Marco Rubio told FOX News any deal would require Iran to dismantle its nuclear program permanently.

MARCO RUBIO: We have to ensure that any deal that is made, any agreement that is made is one that definitively prevents them from sprinting towards a nuclear weapon at any point.

NICK SCHIFRIN: And Rubio appeared to dismiss Iran's willingness to give up its choke hold on the straits.

MARCO RUBIO: And what they mean by opening the straits is, yes, the straits are opened as long as you coordinate with Iran and get our permission or we will blow you up and you pay us.

That's not opening the straits.

Those are international waterways.

They cannot normalize, nor can we tolerate them trying to normalize a system in which the Iranians decide who gets to use an international waterway and how much you have to pay them to use it.

NICK SCHIFRIN: But the fact is, that system did not exist before the war, a fact highlighted today by German Chancellor Friedrich Merz in extraordinary criticism of the U.S.

FRIEDRICH MERZ, German Chancellor (through translator): The Americans clearly have no strategy.

And the problem with conflicts like this is always that you don't just have to go in.

You also have to get out.

An entire nation is being humiliated by the Iranian state leadership.

NICK SCHIFRIN: This weekend, that Iranian leadership's main proxy, Hezbollah and Lebanon, marked the deadliest day since the cease-fire began two weeks ago, 14 dead, some celebrated at this Hezbollah funeral.

Israel is now demanding that residents in at least seven villages north of the Litani River evacuate above the area where Israel has invaded and occupied, leading once again to an exodus of residents from Southern Lebanon.

In that same area, Hezbollah continues to fight.

This weekend, Hezbollah released video of drone attacks on Israeli soldiers.

They killed 19-year-old Idan Fox, buried by his family today.

This sacrifice will continue until Hezbollah can no longer fire rockets or drones, Israeli Prime Minister Benjamin Netanyahu told senior military commanders today.

BENJAMIN NETANYAHU, Israeli Prime Minister (through translator): What I expect from you is to solve these two problems, because I believe we will be able to resolve the diplomatic aspect if we have solved these.

NICK SCHIFRIN: That diplomacy has been between Israel's and Lebanon's ambassadors in Washington.

But, today, Hezbollah accused the Lebanese government of treason during these talks.

Lebanon's president responded, accusing Hezbollah of its own treason for taking Lebanon to war to serve Iranian interests.

Back on the U.S.-Iran negotiations, an Iranian official tells me that Iran is unlikely to adjust its demands, despite what the official told me was the U.S.'

rejection.

And so the official called this moment -- quote - - "totally an impasse."

Iran won't give in to U.S.

nuclear demands, and the U.S.

hasn't been willing to give in to Iranian demands.

And so it's not clear how the two sides can find an off-ramp.

That said, the official told me Iran has been willing to consider a less-than-10-year freeze on domestic nuclear enrichment.

So far, that is not something the U.S.

has said it is willing to accept.