Government forces, led on the ground by Emirati-backed troops, have made their way into Hodeida after 11 days of clashes, reaching residential neighbourhoods in the east on Sunday and sparking fears of street fights that would further endanger civilians trapped in the city.
British Foreign Secretary Jeremy Hunt met with Saudi King Salman on Monday during a visit to the kingdom to press its rulers to support United Nations efforts to end the conflict.
After almost four years of conflict in Yemen, home to what the UN calls the world s worst humanitarian catastrophe, the United States and Britain are now pressing Saudi Arabia and its allies to end the war against the Iran-backed Huthi rebels.
Analysts however say a coalition attack on the docks remains a possibility, which would put at risk 14 million aid-dependant Yemenis already on the verge of mass starvation.
The United States, Britain and France - three of the main arms suppliers to Saudi Arabia - have also called for an end to almost four years of conflict in Yemen, particularly in Hodeida. "That's what the USA says, we're saying and the British too", he told France 2 television.
After almost four years of conflict between Yemen's Iran-linked Huthis and a pro-government military alliance led by Saudi Arabia and the UAE, the fight this month reached rebel-held Hodeida, home to Yemen's lifeline port.
A 15-year-old boy died last week of shrapnel wounds in Hodeida, Save the Children said.
The port's deputy director, Yahya Sharafeddine, said the main entrance to the docks had been "the target of air raids" but was fully functioning.
Four other port employees told AFP that one strike had killed a rebel commander and three of his guards, while a second strike had wounded another commander and his guards.
The coalition has come under intense global pressure to end the conflict in Yemen, particularly following the killing of Saudi journalist Jamal Khashoggi.
After failed peace talks in September, the United Nations is pushing to host a new round of negotiations between the government, backed by the coalition, and the Iran-linked Huthi rebels by the end of the year.
The coalition has been battling Iran-aligned Houthi rebels since March 2015, in a war that has killed tens of thousands of people.
Hodeida port is under a near-total blockade by the coalition, who accuses Iran of smuggling arms to the Houthis. But rights groups believe the toll may be five times as high.