AMNA NAWAZ: Welcome to the "News Hour."

President Trump told "PBS News Hour" today there's a -- quote -- "very good chance" the U.S.

and Iran are close to a deal to end the war and reopen the Strait of Hormuz.

But he added to our Liz Landers that he has felt that confidence before, only for there to be no deal reached.

He echoed that cautious optimism in the Oval Office this afternoon.

DONALD TRUMP, President of the United States: We have had very good talks over the last 24 hours.

And it's very possible that we will make a deal.

We have had some good talks before, as you know, and all of a sudden the next day they're like -- they forgot what happened.

AMNA NAWAZ: The deal on the table would be a negotiated memorandum of understanding that would limit Iran's nuclear program and open the Strait of Hormuz, where Iran has kept its choke hold since the beginning of the war and where U.S.

warships continue to keep a blockade.

Our Nick Schifrin's been reporting out the details of the negotiation.

He's traveling and joins me now from Austin.

So, Nick, what else is in this possible deal and how confident are U.S.

and Iranian officials that they can reach a deal?

NICK SCHIFRIN: Well, as you just mentioned, Amna, President Trump told our colleague Liz Landers that he believes a deal is close, although he's felt that optimism before.

But an Iranian official does confirm to me that there has been progress.

So what's in the deal?

According to an Iranian, a senior regional and a senior European official, they tell me the first phase of the negotiations would focus on the Strait of Hormuz.

Iran would lift its choke hold over the strait and the U.S.

would end its blockade.

All parties would declare an end to the war, including Iranian proxy Hezbollah and Israel.

And then the strait would be opened as quickly as possible.

The second phase would focus on the longer-running issues of contention between the U.S.

and Iran.

Iran would freeze its domestic and uranium enrichment, but the time frame is still under negotiation.

The U.S.

has raised 20 years.

Earlier, Iran countered with five years, with a possible five-year extension.

Today, President Trump told Liz Landers the deal does not include any authorization of low-level Iranian enrichment after that freeze ends, a fact an Iranian official confirmed to me.

In addition, Iran would have to export its nearly 1,000 pounds of highly enriched uranium.

The president told Liz Landers that Iran had to export it to the U.S., although the Iranian official told me it would be exported to the U.N.

's nuclear watchdog.

The IAEA would also gain access to Iranian nuclear sites for inspections.

And the president confirmed to Liz Iran would pledge not to operate underground nuclear facilities, which the Iranian official also confirmed to me.

Now, in exchange for all of that, the U.S.

would lift sanctions on Iran and unfreeze Iranian assets.

The two sides would have about 30 days to negotiate all of those big ideas.

As the president told Liz today -- quote -- "I think it's got a very good chance of ending.

And if it doesn't end, we have to go back to bombing the hell out of them.

Very simple."

AMNA NAWAZ: So, Nick, in all your reporting, how close would you say the two sides actually are to that deal?

NICK SCHIFRIN: Amna, it depends on who you ask.

So we've been talking about the president's cautious optimism.

A senior regional official close to the negotiations told me that they are - - quote -- "the most hopeful" since this started, but there are still -- quote -- "difficult discussions ahead."

And a senior European official who has dealt with Iran extensively over the years told me that they are very doubtful that the two sides can reach this agreement and doubtful that Iran will actually fulfill what it's promising in this agreement.

That said, an Iranian official, Amna, does confirm to me that there has been major progress.

AMNA NAWAZ: Meanwhile, as we've seen before in the region, right before a cease-fire or a possible cease-fire, violence can flare.

There was more fighting today.

Tell us what happened.

NICK SCHIFRIN: Yes, the military's Central Command said that an American jet fired at an Iranian tanker that was trying to run that U.S.

blockade that you mentioned earlier destroyed the rudder on that tanker, disabling it.

But the heaviest fighting today occurred in Lebanon.

Israel launched widespread airstrikes across the south today, where Israel has invaded and occupied.

Israel's top general said today Israeli soldiers were So,the "deepening the dismantling of Hezbollah" and accused Hezbollah of launching drones and rockets that injured multiple soldiers.

The IDF also warned today that it could return to a -- quote -- "powerful and broad campaign" against Iran.

But, Amna, that would require President Trump to give Israel the green light.

And President Trump made clear today that he is at least hoping that this deal can be made quickly.

AMNA NAWAZ: All right, Nick Schifrin, thank you very much.

NICK SCHIFRIN: Thank you.