ERBIL, Kurdistan Region of Iraq – Washington on Tuesday claimed progress toward a possible reopening of the Strait of Hormuz, citing progress in Oman-Iran waterway management talks, while Iran insisted that only it and Muscat have authority over passage arrangements and rejected US interference.

On Monday, US President Donald Trump said talks with Iran were ongoing and that officials were working to reopen the Strait of Hormuz "literally tomorrow." He previously said a potential deal with Iran would include the immediate reopening of the strategic waterway.

The two warring sides have paused hostilities three times to pursue negotiations, but have returned to launching attacks on every occasion.

"There's been progress made in those talks but not finality yet; we're hoping that will happen very shortly," US Secretary of State Marco Rubio told reporters on Tuesday, adding that Washington is "involved" in the Iran-Oman negotiating process.

"I think that there's a conversation and a negotiation that we are involved in between Oman and Iran on how more ships can be able to go through there safely in the short term as we move towards these longer-term talks about denuclearization," he added.

Iranian authorities have repeatedly stressed Tehran's right to maintain control over the vital waterway in cooperation with fellow littoral state Oman and charge fees on vessels in transit, a proposition that Washington has thus rejected, insisting on a greater role in overseeing the waterway and the free passage of ships.

"I think there is a chance we may have a deal today or tomorrow to open the strait and move towards a more normalized position in this conflict," US Treasury Secretary Scott Bessent on Tuesday told CNBC on Tuesday.

"Understanding with Oman on Hormuz is within reach if US stops interference," a senior Iranian official told the state-owned Press TV.

The Iranian official further described the alleged US interference in the talks with Oman as "Trump seeking to claim credit for achievements of Hormuz talks," adding that the US efforts would "only slow negotiations."

Meanwhile, Iran on Sunday said that negotiations with Oman over shipping management in the Strait of Hormuz are entering their final stages, while warning that transit through the strategic waterway "will never return to the pre-war situation."

The dialogue regarding management of the Strait of Hormuz focus on “determining safe return routes for ships; a path that, while ensuring sovereign rights, also takes into account national security considerations of Iran and Oman,” Iranian foreign ministry spokesperson Esmail Baghaei said Tuesday, as quoted by the Iranian state broadcaster IRIB.

"Iran is trying to develop the necessary mechanisms for future arrangements," he said, adding, "The final results of these talks will be announced after they are concluded."

In late June, Tehran and Muscat announced the formation of a joint working group to determine future administrative arrangements for the waterway, including discussions on the “costs associated” with commercial transit through the strait.