AL HUDAYDAH, Yemen — Fighting raged Saturday around the international airport outside the Yemeni port of Al Hudaydah as fighters with a Saudi-led Arab coalition pressed their four-day-old offensive to seize the rebel-held city that is the gateway for food supplies to the famine-stricken country.

Officials loyal to Yemen’s exiled government claimed their forces backed by the coalition had seized the airport, which lies just south of the city, and deployed engineers to clear explosives. But Iran-backed Houthi rebels launched a counteroffensive amid reports of heavy fighting at the airport gates and inside the sprawling compound, which has been closed since 2014.

By Saturday afternoon, warplanes had struck Houthi targets on the edge of the city and fighting spread to a major road leading to the Houthi-held capital, Sana, blocking a key exit from the city. Hundreds of thousands of residents trapped in the city huddled in their homes, many wondering nervously if the fight will reach them.

The battle for Al Hudaydah is shaping up to be the biggest and possibly most consequential of a war that started in 2015 when the Saudi-led coalition attacked the Houthis, who had seized the Yemeni capital a year earlier.