By Wednesday night, the offensive remained on the outskirts of the rebel-held airport.
The Red Sea port, controlled by the Iran-backed Huthi rebels who hail from northern Yemen, serves as the entry point for 70 percent of the impoverished country's imports as it teeters on the brink of famine.
Britain requested an urgent meeting of the United Nations Security Council over the offensive, and the top UN body is expected to hold talks on Thursday.
The request came after the UN envoy for Yemen, Martin Griffiths, said he was still holding negotiations on keeping the key port open to aid deliveries.
"We are in constant contact with all the parties involved to negotiate arrangements for Hodeida that would address political, humanitarian, security concerns of all concerned parties," he said.
The European Union's foreign policy chief warned of the "devastating" impact the assault would have.
"The latest developments will only lead to further escalation and instability in Yemen, thus undermining the ongoing efforts of the UN special envoy," Federica Mogherini said in a statement.
In Hodeida, people waited anxiously for the fighting to reach their neighbourhoods. Those contacted by AFP said Huthi fighters had fanned out across the city.
Coalition sources said the alliance carried out 18 air strikes on Huthi positions on the outskirts of Hodeida on Wednesday.
According to medical sources in the province, 22 Huthi fighters were killed by coalition raids, while three pro-government fighters were killed in a rebel ambush south of Hodeida.
The city, home to 600,000 people, was captured by the insurgents in 2014 along with the capital Sanaa.
Saudi Arabia, the United Arab Emirates and a bloc of other countries intervened in Yemen the following year with the goal of restoring the government to power.
The coalition accuses the Huthis of using the port to secure Iranian arms, notably ballistic missiles the militants have increasingly fired into Saudi territory.
Yemen's government said Tuesday that negotiations had failed to force the rebels from Hodeida, and that a grace period for UN-led peace efforts was over.
The UAE, a pillar of the anti-Huthi coalition, says crucial aid will not be halted by the offensive.
"Hodeida port remains open to shipping," Emirati State Minister for Foreign Affairs Anwar Gargash tweeted Wednesday.
"We have also put contingency plans in place to move aid by other methods to Hodeida and points beyond," he said, echoing a statement by UAE Minister of State for International Cooperation Reem al-Hashemi.
The Emirati diplomats insisted that similar large-scale offensives had improved the lives of Yemenis elsewhere.
Hassan Taher, the official Hodeida governor based outside the city, said Wednesday that Yemen's government would ensure aid deliveries throughout the offensive.
"We want a rapid war, but it won't be finished in two days," Taher told AFP.
The International Committee of the Red Cross said any battle for Hodeida would exacerbate the suffering of the population.
"Life-saving items cannot be given to those in need while fighting is ongoing," ICRC Middle East director Robert Mardini said in a statement.
Children in the crossfire
The UN children's fund, UNICEF, has raised alarm over the plight of Hodeida's 300,000 children and the risk that drinking water supplies will be disrupted.
"UNICEF has pre-prepositioned supplies in Hodeida: over 20,000 basic hygiene kits ... We hope we don't need to use them," its Yemen representative Meritxell Relano said on Twitter.
The Huthi leadership on Tuesday called on the international community to "pressure a halt to the escalation", warning an assault on Hodeida would put Red Sea navigation at risk.
On Wednesday, the Huthis said they targeted a coalition warship off the coast of Hodeida with two missiles, with rebel outlet Al-Masirah claiming a direct hit.
As the offensive got under way, Saudi Arabia's ambassador to Yemen was defiant.
"Hodeida will be liberated, and the Yemeni people will gain back a major artery of life," he tweeted.