Yemen's Houthi rebels have threatened to block the vital shipping route that connects the Mediterranean Sea with the Arabian Sea and Indian Ocean.
The president of the so-called supreme political council, which runs Sanaa and other rebel-held territory, made the threat in a meeting with the deputy UN envoy to Yemen on Monday, according to both Saudi and Houthi media.
Council president Saleh Al Samad told the UN's Maeen Shureim that the Houthis will turn to "strategic choices" if the Saudi-led coalition fighting in Yemen does not halt efforts to retake ports in Hodeidah province, where coastline runs along the Red Sea.
Pro-government forces — backed by the coalition — have been making rapid advances in rebel-held Hodeidah, hoping to prevent the Houthis from using the province's ports to smuggle in weapons from Iran.
According to Saudi Arabia's Al Hadath news channel, Mr Al Samad said the rebels would "cut international navigation", while the Houthis' Al Masirah TV elaborated, reporting that he threatened to "block the Red Sea and target international navigation".
Dr Anwar Gargash, the Minister of State for Foreign Affairs, responded on Tuesday, tweeting that the threats were "further affirmation of the terrorist nature of the Houthi militias".
The Red Sea runs between two chokepoints, the Suez Canal and the Bab El Mandeb.
The US Energy Information Administration estimated that 4.7 million barrels of oil were transported through the latter daily in 2014, mostly headed to markets in Europe.
On the other side of the Suez Canal lies the Mediterranean Sea and on the other side of the Bab El Mandeb the Gulf of Aden. Beyond that is the Arabian Sea and Indian Ocean.
Mr Shureim arrived in Sanaa on Saturday to resume peace negotiations. He was due to meet with Houthi leaders to discuss paving the way for another round of negotiations with the government.
In his meeting with Mr Shureim on Monday, Mr Al Samad criticised the United Nations' efforts to resolve the war in Yemen, including the role of special envoy Ismail Ould Cheikh Ahmed, Al Masirah TV reported.
"We've come to a stage where we don't care any more about the role of the UN in solving the crisis in Yemen," Mr Al Samad was quoted as saying by Saudi news channel Al Arabiya.
The Saudi-led coalition, in which the UAE plays a leading role, has been fighting in Yemen since March 2015. It wants to restore president Abdrabu Mansur Hadi to power. His internationally recognised government was forced to flee the capital, Sanaa, after it was seized by the Houthis in September 2014.