US President Donald Trump said on Wednesday that Iran has told the United States that no tolls were being sought from ships traveling through the Strait of Hormuz.

The two countries, which ended ‌a first round of negotiations in Switzerland on Monday, have offered conflicting accounts about financial incentives for Iran, control of the Strait of Hormuz, and Israel's parallel war in Lebanon - all major aspects of their framework deal signed last week, aiming to end the war.

Trump has faced criticism over the deal domestically, including from hardliners in his Republican Party.

"Iran has informed the US that, despite troublemaking Fake News reporting to the contrary, there are 'NO TOLLS, NO INSURANCE COSTS, & NO OTHER CHARGES OF ANY KIND BEING SOUGHT OR RECEIVED BY IRAN ON SHIPS TRAVELING THE STRAIT OF HORMUZ' Trump wrote in a social media post.

"If this is false information, negotiations would end, immediately!"

Ships have begun sailing through the Strait of Hormuz under a new scheme by the UN's shipping agency to evacuate vessels trapped there by the conflict, a spokesperson said on Wednesday.

The initiative, which has taken months to conclude, will enable hundreds of ships with some 11,000 seafarers stranded in the Gulf to sail through Hormuz, the International Maritime Organization said on Tuesday.

"Ships have already begun to pass under the plan," an IMO spokesperson said on Wednesday, declining to provide any details of the vessels that had crossed.

At least two dry bulk ships and one cargo ship have sailed through Hormuz under the scheme in the past 12 hours, LSEG ship tracking data showed on Wednesday.

At least 35 other commercial ships, mainly dry bulk, cargo, and container vessels, were preparing to sail through the strait, according to LSEG and MarineTraffic ship tracking data based on Reuters analysis of ship movements.

Those vessels are smaller commercial ships, including five smaller oil tankers, coastal ships, and tugs, according to an analysis of the vessels waiting.

Under the scheme, which the IMO said was able to begin after the US and Iran reached a ceasefire framework, vessels will be able to use two tracks to sail out - a northern route via Iranian waters and a southern route via "the Sultanate of Oman/United States-coordinated waters."

"Vessels should wait for instructions before proceeding," the IMO said in a note on the scheme issued on Wednesday.

"Crowding the waiting area will only result in the need to pause further notifications for the safety of navigation."

In recent weeks, the US military has launched a mission to help ships out of the strait.