US Secretary of State Marco Rubio said on Friday that the United States expected a response within hours.
But a day later, there was no sign of movement from Iran on the proposal, which would formally end the war before talks on more contentious issues, including Iran's nuclear program.
A Qatari liquefied natural gas tanker was sailing toward the strait on Saturday en route to Pakistan, according to LSEG shipping data, in a move sources said was approved by Iran to build confidence with Qatar and with Pakistan, a mediator in the war.
If completed, it would mark the first transit of a Qatari LNG vessel through the strait since the conflict started.
With US President Donald Trump due to begin a visit to China next week, there has been mounting pressure to draw a line under the war, which has thrown energy markets into turmoil and posed a growing threat to the world economy.
Recent days have featured the biggest flare-ups in fighting in and around the strait since a ceasefire began a month ago, and the United Arab Emirates came under renewed attack on Friday.
On Friday, there were sporadic clashes between Iranian forces and US vessels in the strait, the Iranian Fars news agency reported.
The Tasnim news agency later cited an Iranian military source saying the situation had calmed but warning more clashes were possible.
The US military said it struck two Iran-linked vessels attempting to enter an Iranian port, with a US fighter jet hitting their smokestacks and forcing them to turn back.
The UAE said its air defences engaged with two ballistic missiles and three drones from Iran on Friday, with three people sustaining moderate injuries.
Iran has repeatedly targeted the UAE and other Gulf states that host US military bases.
In what the UAE called a major escalation, Iran stepped up attacks this week in response to Trump's announcement of "Project Freedom" to escort ships in the strait, which he paused after 48 hours.
Trump said on Thursday the ceasefire, announced on April 7, was holding despite the flare-ups while Iran accused the US of breaching it.
"Every time a diplomatic solution is on the table, the US opts for a reckless military adventure," Iranian Foreign Minister Abbas Araqchi said on Friday.
