The intervention, if confirmed, by the exiled Ahmed Ali Saleh could shift the balance of power yet again after a dramatic week during which the elder Saleh abandoned his Houthi allies.
They responded by killing him and routing his family's forces from the capital Sanaa.
Saleh's funeral is expected later on Tuesday.
What did Ahmed Ali Saleh say?
It was not immediately possible to verify the authenticity of the report of his comments. The younger Saleh, a former leader of the elite Republican Guards, has been exiled in the United Arab Emirates, a country that backs the Saudi-led coalition.
"I will lead the battle until the last Houthi is thrown out of Yemen...the blood of my father will be hell ringing in the ears of Iran," Ahmed Ali Saleh is quoted as saying by the Saudi-owned al-Ekbariya TV.
He also reportedly called for his father's backers to "take back Yemen from the Iranian Houthi militias".
Political sources say Ahmed Ali has been held incommunicado and under guard at a villa in the UAE capital, Abu Dhabi.
His reported first public statement may indicate that his former enemies in the coalition are intending to unleash him against the Houthis.
He appears to have been groomed to succeed his father and may be the family's last chance to win back influence. The whereabouts of Saleh's other key relatives, who had led six days of street battles against the Houthis in the capital Sanaa before being routed on Monday, were unknown.
What the Houthis have said
Their leader, Abdul Malik al-Houthi, hailed Saleh's death in a speech on Monday as a victory against a treasonous conspiracy by Yemen's Saudi enemies.
He also reached out to Saleh's political party, saying his movement has no quarrel with it.
Why are the United Arab Emirates involved?
The UAE is a key member of the mostly-Gulf Arab alliance that sees the Houthis as a proxy of their arch-enemy, Iran. The Gulf countries had struggled to make gains against the Houthi-Saleh alliance, despite thousands of air strikes backed by Western arms and intelligence.
They have used their air and sea power to tightly restrict imports, action the UN says could lead to mass hunger.
What is everyone else saying?
The Arab League’s general secretariat condemned the Iran-aligned Houthi movement which killed Saleh as a “terrorist organization” and demanded that the international community view it as such.
Quiet in Sanaa
Yemen's capital was quiet on Tuesday after five days of fighting and 25 airstrikes overnight.
Residents reported that fighting had subsided but Saudi-led coalition jets pounded several targets, including the downtown presidential palace where the governing body led by Houthi-Saleh politicians had regularly convened.
The UN's humanitarian coordinator, Jamie McGoldrick, said UN and Red Cross aid flights had landed at the airport.
In the southern city of Aden, where the Saudi-backed government is based, residents set off fireworks and expressed joy. Saleh was almost universally hated throughout southern Yemen after he launched a war to unify the country in 1994.
The humanitarian crisis
Yemen's civil war, pitting the Iran-allied Houthis who control Sanaa against a Saudi-led military alliance backing a government based in the south, has led to one of the world's worst humanitarian crises.
The United Nations is warning of a potential famine that could threaten the lives of millions.
"Dancing on the heads of snakes"
The death of Saleh, who once compared ruling Yemen to "dancing on the heads of snakes" deepens the complexity of the multi-sided war.
Much is likely to depend on the future allegiances of his loyalists, who had helped the Houthis, a Shi'ite insurgent movement, seize and hold much of the country until he dramatically switched sides on Saturday.
Arab states, which mainly support the Saudi-backed government, condemned the killing of the veteran ex-leader, saying his death could cause an “explosion” in the country.
What is Saleh's legacy?
A mixed one. He is still loved in much of the north. Many of his supporters will bear a grudge towards his killers.
He ruled in Sanaa from 1978-2012 and had a strong following in the country, including army officers and armed tribal leaders who once served under him.
His supporters may still be able to have some impact on the war.
Some fear Saleh's death may create more instability in Yemen.
"We expect things will get worse for us. This will be the beginning of a new conflict and more bloodshed. The war will not end soon," said Aswan Abdul Khalid, an academic from the University of Aden.
Why is this happening?
The Arabian peninsula's poorest country, Yemen is one of the most violent fronts in a proxy war between Saudi Arabia and Iran.
They have also backed opposing sides in Syria, Iraq and elsewhere across the Middle East.
The Saudi-led coalition had been counting on Saleh's decision to switch sides to tip the balance of the conflict that has been stalemated on the battlefield.