A senior military official with Yemen's internationally recognised government said the Iran-backed Houthi rebels have taken the island of Mayun after government forces withdrew from there on Thursday.
A Houthi official confirmed the rebels were on the island, also known as Perim, about 3.5 kilometres off Yemen's west coast.
Both officials spoke on condition of anonymity because they were not authorised to talk to journalists.
The strait is a key alternative to the Iran-choked Strait of Hormuz, and the Houthis say their advance is part of an effort to target Saudi shipping.