“Only small pockets remain,” Hussain Albukhaiti, who has close ties to the Iranian-backed Houthis, said on Monday.
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Albukhaiti spoke to Al Jazeera before the news of Saleh’s death. The former Yemen president was killed later on Monday in fighting in Sanaa.
Albukhaiti said that fighters had secured key areas south of the capital, including the “very strategic” Al-Mesbahi residential area, which is approximately 200 metres from Saleh’s home.
“The area around his home is completely surrounded and may be taken over by the Houthis within the next few hours,” he said.
Other key places that have been taken over by the Houthis include a building where Saleh’s son, Ahmed Abdullah Saleh, resides, and the local al-Saleh mosque.
As clashes continued between the two former allies for the fifth, consecutive day, Albukhaiti noted that infighting was centred on one street.
On Sunday night, the Saudi-led coalition launched air strikes in and around Sanaa, targeting the capital’s abandoned airport and the ministry of interior, said Albukhaiti.
Earlier this week, Saleh had expressed his openness to talks with the Saudi-led coalition fighting the Houthi rebels, in what the fighters called “a coup” against their fragile alliance.
Saleh made the comments on Saturday, as deadly infighting between forces loyal to him and Houthi rebels continued.
The Saudi-led coalition praised Saleh for “taking the lead” in the conflict.
The former allies have been fighting the Saudi-led coalition for control of the country since March 2015, when Riyadh and several other Arab Sunni states intervened to reinstate the government of President Abd-Rabbu Mansour Hadi.
According to Albukhaiti, coalition air strikes have been targeting areas that are now under Houthi control.
He noted that the number of those who died and those suffering injuries surpasses 100 people – including fighters on both sides as well as civilians.
The spokesperson for the United Nations Secretary-General voiced his concern over the “sharp escalation of armed clashes and air strikes” in Sanaa.
In a statement on Sunday, Stephane Dujarric said that the fighting has already resulted in “dozens of deaths”.
“Fighting is restricting the movement of people and life-saving services within Sanaa city. Ambulances and medical teams cannot access the injured and people cannot go outside to buy food and other necessities,” he said. “The Secretary-General calls on all parties to the conflict to cease all air and ground assaults.”
Infighting comes as the residents of the partially blockaded city are in dire need of humanitarian assistance.
The Saudi-led coalition imposed a blockade in October on the country, where nearly 80 percent of residents need humanitarian aid to survive.
Last week, amid mounting international pressure over the suffering of millions of Yemenis, some humanitarian aid was allowed to enter the country.
The International Committee of the Red Cross’ regional director, Robert Mardini, said in a Twitter post that the aid organisation’s “medical warehouse was hit”.
Mardini also said civilians were frightened and unable to leave their homes.
The war in Yemen is one of the worst humanitarian crises in the world, killing at least 10,000 people and leading to widespread hunger and disease.
The night was tough in #Sanaa, #Yemen. Massive urban clashes with heavy artillery & airstrikes. #Yemenis stuck in their homes, too scared to go out. Reduced access to water, health care, food & fuel. Increased vulnerability of #Yemenis put in harm's way.
— Robert Mardini (@RMardiniICRC) December 4, 2017