Iran fired cruise missiles at the USS Abraham Lincoln aircraft carrier, the military said in a statement carried by state television, warning of further launches when the strike group's ships come in range.

Israel's Channel 12 on Tuesday published a 15-point U.S. document that is being reportedly developed to halt the war with Iran, amid Israeli concerns that the agreement could lead to uncertain outcomes.

U.S. President Donald Trump said Tuesday that Iran gave him a "very big present" related to the Strait of Hormuz, boosting his confidence that he was talking to the right people in Tehran to end the war.

The Israeli military said on Tuesday that it was operating "according to an unchanged plan" in Iran and Lebanon, regardless of any talks aimed at reaching an agreement to end the hostilities.

The U.S. and a group of regional mediators are discussing the possibility of holding high-level peace talks with Iran as soon as Thursday, but are still waiting for a response from Tehran, two sources with knowledge of the discussions told U.S. news portal Axios.

Long before he became Iran's parliament speaker, Mohammad Bagher Qalibaf went on a charm offensive for almost two decades, portraying himself as a hard-liner the West could do business with in the Islamic Republic.

"I would like the West to change its attitude to Iran and trust Iran, and rest assured that there's an attitude in Iran to advance issues through dialogue," he told The Times newspaper of London in 2008.

Iran and Israel traded strikes on Tuesday, as the Middle East war showed no sign of de-escalation despite U.S. President Donald Trump signaling "very good talks" to end the three-week conflict.

The war, sparked by U.S.-Israel attacks on Iran that killed its supreme leader, has upended global energy markets, roiled the world economy, and spiraled throughout the region -- even dragging in safe-haven Gulf nations.

Israeli Prime Minister Benjamin Netanyahu said Monday that he spoke with Donald Trump, saying that the U.S. president believed U.S.-Israeli military gains in Iran could be converted into a negotiated agreement that protects Israel's interests.

"President Trump believes there is a chance to leverage the tremendous achievements of the IDF (Israeli army) and the U.S. military in order to realize the war's objectives in an agreement -- an agreement that will safeguard our vital interests," Netanyahu said in a video statement.

U.S. President Donald Trump told reporters on Monday that his envoys held talks with a senior official in the Iranian leadership and claimed the parties are in agreement on many points.

Trump walked back his threat to strike Iran's power plants on Monday, citing productive negotiations.

President Donald Trump said Monday that there are "major points of agreement" in U.S.-Iran talks which he said must result in Tehran giving up its nuclear ambitions and enriched uranium stockpile.