This crucial sea route, which Iran has effectively closed in response to US-Israeli attacks, is located in the territorial waters of both sides.
Negotiations have been going on for two weeks.
An Omani source told CNN that negotiations have been ongoing for two weeks and are aimed at establishing a "safe passage protocol" that would allow commercial shipping access.
Under the proposal, oil and gas tankers passing through the narrow waterway, which has been de facto blocked since US and Israeli forces attacked Iran last month, would have to register with Iran and agree to sail under the flag of a country not considered hostile, a source told CNN - effectively acknowledging Iranian control over the strategic route.
The proposals appear to be part of a latest attempt by energy-rich Oman, a country that has previously mediated between Washington and Tehran, to prevent a wider escalation of the war.
But the same source told CNN that the proposed deal to open the Strait of Hormuz "would be difficult to achieve in wartime conditions," adding that "the best solution remains an immediate ceasefire as a first step toward a comprehensive end to the conflict."
"Escalation is a new risk"
A source has described US President Donald Trump's threat to step up attacks on Iran after a new 48-hour deadline expires on Monday morning as "alarming".
"Escalation carries the risk of untold suffering in many parts of the world," the source told CNN.
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