Loss by the Houthis of the waterway could not be confirmed, the waterways connects the Red Sea with the Gulf of Aden and the rest of the Indian Ocean.
"In a large-scale military operation launched today, Yemeni government, resistance and coalition forces liberated the Bab al-Mandeb strait and Mayun island with the goal of safeguarding this key sea route," Rajeh Badi told Reuters by telephone from the southern port city of Aden.
Bloom Gist learnt from sources that the air strikes and shelling by warships in support of a ground push towards the area.
However, they could not immediately confirm that the strait had fallen.
A Yemeni military official who spoke to DPA news agency on Thursday said there were casualties on both sides in the battle.
Bab al-Mandeb forms part of the world's busiest shipping routes, with much of the Middle East's oil exports passing through the body of water.
More than 3.4 million barrels of oil per day passed through the Bab al-Mandeb in 2013, according to the US Energy Information Administration.
Source: Agencies
