After a breakdown of weekend talks in Islamabad between the two adversaries, a US official said there was continued engagement and forward motion on trying to get to an agreement.
Pakistani Prime Minister Shehbaz Sharif also said efforts were still under way to resolve the conflict.
US President Donald Trump said Iran had been in touch on Monday and wanted to make a deal but that he would not sanction any agreement allowing Tehran to have a nuclear weapon.
Since the United States and Israel began the war on February 28, Iran effectively shut the Strait of Hormuz to all vessels except its own, saying passage would be permitted only under Iranian control and subject to a fee.
The fallout has been widespread, since nearly a fifth of the world's oil and gas supplies flowed through the narrow waterway before the start of the conflict.
Trump has said Washington would block Iranian vessels and any ships that paid such tolls and that any Iranian "fast-attack" ships that went near the blockade would be eliminated.
Tehran has threatened to hit naval ships going through the strait and to retaliate against its Gulf neighbours' ports.
Shipping data on LSEG showed Chinese-owned oil-and-chemicals tanker Rich Starry passed through the strait on Tuesday - the first since the US blockade began at 10am, Washington time, on Monday (12am on Tuesday AEST).
The vessel, which departed Sharjah anchorage off the coast of Dubai on Monday heading for China, had earlier turned back minutes after approaching the strait.
The US's blockade has further clouded the outlook for global energy security and the supply of a vast array of goods that relies on petroleum, and has little, if any, international backing.
NATO allies including Britain and France said they would not be drawn into the conflict by taking part in the blockade, stressing instead the need to reopen the waterway.
Despite the breakdown of talks between the US and Iran on Sunday, Vice-President JD Vance, who led the US delegation, told Fox News on Monday the US "made a lot of progress" by communicating to Tehran where the US "could make some accommodation" and where it would remain inflexible.
He said Trump was adamant that any enriched nuclear material must be removed from Iran and a mechanism must be established to verify that Iran was not developing nuclear weapons.
Tehran "moved in our direction, which is why I think we would say that we had some good signs, but they didn't move far enough", Vance said.
The ceasefire that halted six weeks of US-Israeli air strikes and retaliatory fire from Iran across the Gulf looked in jeopardy, with only a week left to run.
The US military's Central Command said the blockade would be "enforced impartially against vessels of all nations" entering or leaving Iranian ports in the Gulf and Gulf of Oman.
It would not impede neutral transit through the Strait of Hormuz to or from non-Iranian destinations.
The Iranian military called any US restrictions on international shipping "piracy", warning that if Iranian ports were threatened, no port in the Gulf or Gulf of Oman would be secure.
Any military vessels approaching the strait would violate the ceasefire, Iran's Revolutionary Guards said.
Israel has continued to bombard Lebanon and on Monday troops launched an attack it said was intended to seize a key south Lebanon town from Iran-backed Hezbollah.
Israel and the US have said the campaign against Hezbollah was not part of the ceasefire, while Iran has insisted it is.
