The United States said on Monday that it was still close to reaching a "solid" agreement with Iran, after Donald Trump the previous day tempered hopes of an imminent agreement to bring a lasting end to the war in the Middle East.

"We have what I believe is a pretty solid thing on the table regarding their ability to open the Strait of Hormuz, but also to enter into negotiations on Iran's nuclear program," said US Secretary of State Marco Rubio from New Delhi.

"We thought we would have news last night, maybe today (Monday), I wouldn't go too far on that," he added.

Triggered on February 28 by an American-Israeli attack on Iran, the conflict has spread to a large part of the Middle East and has caused thousands of deaths, especially in Iran and Lebanon where the pro-Iranian movement Hezbollah joined the hostilities in early March by targeting Israeli territory.

A ceasefire has been in effect since April 8 between Iran and the United States, but the global economy continues to be shaken by the near-blockade of the strategic Strait of Hormuz initiated by Iran for almost three months.

While the United States works to reach an agreement, President Donald Trump tempered hopes on Sunday, despite signs of progress on both sides.

"I have asked my representatives not to rush into an agreement, because time is on our side," he wrote on his Truth Social platform, also warning that the blockade imposed by his country on Iranian ports would remain in effect "until an agreement is reached, certified and signed."

The presidency believes, however, "that approval of the agreement by the Iranian authorities could take several days," according to the media outlet Axios.

"As the president said, he is not in a hurry, he is not going to make a bad deal, and the president will not sign a bad deal," Marco Rubio stressed on Monday.

According to US media, the agreement being prepared between Tehran and Washington would allow ships to once again pass through the Strait of Hormuz, through which a fifth of the world's hydrocarbons were consumed before the conflict.

Fueled by these hopes of an agreement, oil prices retreated Monday morning in Asia. Prices for Brent crude from the North Sea and West Texas Intermediate (WTI) crude fell by more than 5% shortly after 04:00 GMT.

What about nuclear power?

According to CBS News, citing sources close to the discussions, the latest proposal would also include the unfreezing of certain Iranian assets in banks abroad.

"Despite discussions that began today (Sunday), the United States continues to block certain clauses of the agreement, notably the issue of unfreezing Iranian assets, and these points remain unresolved at this time," the Tasnim news agency reported at the end of the day.

Fars, for its part, reports that sanctions targeting oil, gas and other petrochemical products would also be lifted while negotiations continue in order to allow Iran to export these products, which are essential for its economy.

The agreement under discussion, however, does not appear to resolve the nuclear issue.

"Nuclear negotiations are highly technical issues. You can't settle a nuclear issue in 72 hours over a napkin," Marco Rubio told the New York Times.

Once the reopening of the Strait of Hormuz is achieved, "we will begin, according to agreed terms, very serious negotiations on enrichment, on highly enriched uranium and on their commitment never to acquire nuclear weapons," he said, mentioning in this regard a period of "60 days".

Israeli Prime Minister Benjamin Netanyahu said Sunday he and Mr. Trump agreed that any final deal with Iran must "completely eliminate the nuclear threat," according to a statement after a phone call Saturday night between the two allies.

Pakistani Prime Minister Shehbaz Sharif, mediator in the negotiations, fueled speculation on Sunday about a phased resolution to the conflict, stating his hope to "host the next round of talks very soon." The first round of negotiations was held in Islamabad on April 11, but it failed to produce any results.

"Right to defend oneself"

On the Lebanese front, two people were killed Sunday by Israeli strikes, the day after an Israeli raid that killed 11, the Ministry of Health announced, despite the truce in place since April 17.

And the Israeli army announced on Monday the death of one of its soldiers the previous day in southern Lebanon.

Mr. Netanyahu stated on Sunday that Donald Trump, during their phone call, reiterated Israel's "right" to defend itself on all fronts, including in Lebanon. "Israel always has the right to defend itself. Every country in the world has that right," Marco Rubio added on Monday.

Hezbollah leader Naim Qassem said he hoped the agreement between Washington and Tehran would include his country. But the Shiite movement's leader also reiterated his rejection of direct negotiations between the Lebanese government and Israel, a fourth round of which is scheduled for early June in Washington, and repeated that the disarmament of his organization, demanded by the Lebanese authorities, would be the realization of an "Israeli project."