World leaders are becoming increasingly concerned that the strait’s closure is causing a cost-of-living crisis.
The European Union said this week that it would release emergency funds to help farmers deal with the soaring costs of fertiliser trapped in the Gulf.
The proposal for a Nato mission to unblock the waterway was first reported by the Bloomberg news agency.
It was said to have the support of a small number of Nato allies but will need unanimous approval to proceed.
Levels of support could grow the longer the strait remains blocked, one diplomat said.
General Alexus Grynkewich, Nato’s Supreme Allied Commander Europe, said he was considering how the alliance could contribute.
He said: “Am I thinking about it? Absolutely. But there’s no planning yet until the political decision is taken.”
With discussions at an early stage, it is unclear how Nato would proceed to unblock the waterway. Many of its allies are reluctant to move before a ceasefire because it would drag them into a wider war.
The Anglo-French coalition has already pre-positioned a number of warships and other assets to the region. In effect, Nato commanders could take over the operation, given that many of its backers are part of the alliance.
But accommodations would have to be made for Spain, which has banned the US from using its military bases and airspace to attack Iran.
Other countries have also been critical of Trump’s war but have quietly permitted use of their facilities to provide logistical support.
Europe’s initial refusal to help unblock the strait became a point of tension between Nato and Trump, after he demanded they assist in the region.
Trump’s criticism of the alliance has sparked fears that he will refuse to appear at its annual summit scheduled to take place in Ankara, Turkey, in early July.
Any fresh offer for Nato to intervene in the Middle East could be seen as an olive branch to Trump in the hope that he attends to sign off on the mission.
It is unclear whether it will be enough to win him over.
Meanwhile, Pakistani officials have been attempting to keep negotiations between Washington and Tehran alive after a number of setbacks.
But Iran’s negotiators are refusing to give up on the right to enrich uranium in defiance of Trump’s core demands to curtail its nuclear programme.
Kazem Gharibabadi, Iran’s deputy foreign minister, was also reported as saying Tehran was demanding the easing of sanctions, the release of frozen funds and the end of the US’s blockade of the strait.
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