The claim, which could not be independently confirmed, came as Yemen's long and devastating civil war took a potentially decisive turn.
Heavy clashes were reported in the city between forces loyal to Saleh and the Iran-backed Shiite rebels, who together seized control of Sanaa from the internationally recognised government three years ago.
Moving to take advantage of the chaos, President Abedrabbo Mansour Hadi, whose government is backed by a Saudi- led military coalition, ordered his forces to advance on the capital.
A statement read on the channel announced the "end of the crisis of militias", referring to Saleh's armed supporters, and "the killing of their leader and a number of his criminal supporters".
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A video given to AFP by the rebels showed what appeared to be a dead Saleh with a severe head injury, his body wrapped in a floral-print blanket.
An AFP photographer who approached Saleh's home in southern Sanaa today found it in the hands of the Huthis and was prevented from entering. The house appeared to have been damaged in fighting.
As witnesses reported continued heavy fighting today, Hadi ordered his forces to launch an offensive to advance on the capital.
"The president has ordered Vice President Ali Mohsen al- Ahmar, who is in Marib (east of Sanaa), to activate military units and advance towards the capital," a presidency official told AFP, speaking on condition of anonymity.
The government also reached out to Saleh's supporters with an offer of amnesty.
The Saleh-Huthi alliance had been fraught since its inception in 2014, when the two ended decades of enmity and joined ranks to capture Sanaa from Hadi's government.
Saudi Arabia, accusing arch-rival Iran of backing the rebels, intervened in the Yemen war on behalf of the government the following year.
The Saudi-led coalition has been hitting Sanaa with air strikes for months, and a fresh wave today targeted areas near Sanaa International Airport and the interior ministry, both under Huthi control, residents and an airport source said.
"The coalition urges civilians to evacuate areas near positions held by the Huthis," read a statement published by Saudi Arabia's state-run Al-Ekhbariya TV.
"We ask civilians to remain at least 500 metres away from Huthi military vehicles and gatherings."
Residents near the airport said multiple air raids had shaken their homes late on Sunday night and early today.
An airport source said rebel bases near the location appeared to have been targeted, but the airport itself had not been bombed.
Residents reported that the fighting, which erupted Wednesday, had spread outside the capital.
Witnesses said clashes had erupted around the residence of Tarek Saleh, a nephew of the former president and a leader with his forces.
Saleh on Saturday announced he was open to talks with Saudi Arabia and its allies on condition they ended their crippling blockade of Yemen's ports and airports.
That dealt a serious blow to his already fragile alliance with rebel chief Abdul Malik al-Huthi.
The Saleh-Huthi split sparked fears of a new front in the Yemen war, which has already claimed more than 8,750 lives since the Saudi-led coalition intervened.
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