Several houses in the central Crater district were set on fire after being hit by rockets as the rebels advanced, residents told the Reuters news agency.
Warplanes of the Saudi-led coalition, which backs the government, meanwhile bombed rebel targets to the north.
Iran is also reported to have sent navy vessels to the Gulf of Aden.
Navy commander Rear Adm Habibollah Sayyari was quoted as saying by state-run Press TV that a destroyer and another ship would be deployed off Yemen as part of an anti-piracy campaign “safeguarding naval routes for vessels in the region”.
Tehran has denied Saudi accusations that it is providing military and financial assistance to the Houthis, who adhere to a branch of Shia Islam known as Zaidism.
The campaign by Sunni-ruled Saudi Arabia, which seeks to “defend the legitimate government” of Mr Hadi, has so far failed to stop the Houthis’ assault on Aden.
Humanitarian organisations have struggled to treat those wounded by the fighting in Aden,
The president took refuge in the second city in February after fleeing the capital, Sanaa, where he had been held under house arrest by the rebels. When the Houthis reached the outskirts of Aden on March 25, he left the country.
Yesterday, dozens of rebel fighters and allied troops reportedly pushed into the district of Crater, near the city’s port. Residents told Reuters that the rebels were backed by a tank and two armoured vehicles.
They also said that from loudspeakers on Sunni mosques a call had rung out for local people to “rise for jihad” against the attackers.