The Houthi rebels and their allies made a new push on a port in the central Mualla district of the city but were forced back by militia loyal to fugitive President Abd-Rabbu Mansour Hadi, witnesses said.
Naval forces of the Saudi-led coalition, which has carried out nearly two weeks of air strikes in support of Mr Hadi, shelled rebel positions across the city, they added.
Spokeswoman for the International Committee of the Red Cross in Yemen, Marie Claire Feghali, said the humanitarian situation across the country was “very difficult… (with) naval, air and ground routes cut off”. The situation in Aden was “catastrophic to say the least”. “The war in Aden is on every street, in every corner… Many are unable to escape,” she said.
Doctors Without Borders (MSF) said the situation was “worsening by the day”.
Medics in Aden had “not received large numbers of casualties over the past few days… due to the difficulties faced in trying to reach a hospital”, said MSF’s Marie-Elisabeth Ingres.
MSF has a team of 140 local staff and eight expatriates at a hospital in Aden. “Our priority is to find a way to send a supporting medical team,” Ingres said, adding a team was waiting in Djibouti “for a green light from the coalition”.
General Ahmed Assiri, spokesman for the coalition, said later permits had been issued for a boat carrying aid and medics from Djibouti.
The Red Cross hopes to deliver to Sanaa on Wednesday 16 tonnes of medical aid on a plane loaded in Jordan. Another plane carrying twice as much could follow the next day.
At least eight Houthis were killed in a coalition strike north of Aden, a military source said.
Other raids targeted air defence posts in the northeast of Taez province, as well as Al-Sadrayn military base in the north of Daleh province.
Meanwhile, 10 Houthis and three tribesmen were killed in Shabwa, said tribal sources.
Overnight fighting in Aden left at least 10 people dead. That was on top of at least 53 people killed in the previous 24 hours.
Nationwide, more than 540 people have been killed and 1,700 wounded since March 19, the World Health Organisation said.
Overnight, Saudi-led warplanes struck the rebel-held Al-Anad air base north of Aden, a general said.
Further east, Al Qaeda’s Yemen franchise sought to tighten its grip on Hadramawt province.
Loud explosions were heard as the militants attacked an army base in the provincial capital Mukalla, much of which they captured last week.
Observers have warned Al Qaeda could exploit the fighting to expand its control following the withdrawal of US troops overseeing a longstanding drone war against it.