Saleh's calls came during a live television address on Saturday, in which he vowed to change if Saudi Arabia stopped its "aggression" and lifted restrictions on Yemen, calling for "opening a new page with neighboring countries".
After the clashes in August, both sides tried to calm the situation because they usually fight on the same side against the pro-government forces and the Saudi-led coalition, so things between them were quiet until Wednesday.
The two groups are fighting a Saudi-led coalition that has intervened in a 2-1/2 year Yemeni civil war with a view to restoring the internationally recognized government of President Abd-Rabbu Mansour Hadi.
Once allies, rebel Houthi fighters and forces loyal to former President Ali Abdullah Saleh have now turned against each other.
Saudi Arabia has led coalition of Sunni Arab nations that have been fighting Saleh and the Iranian-backed Houthi rebels since 2015.
Mr Saleh, who led Yemen for more than 30 years, was deposed in a 2011 uprising. There has been no official word on casualties but the International Committee of the Red Cross (ICRC) said that dozens were killed and hundreds were wounded in the fighting. The country has since fallen into chaos and Mr Saleh later joined the Houthis to drive Mr Hadi out of the capital in 2014. The forces expelled Houthis from Mahwit city and were also able to control the defense ministry complex in Sanaa.
According to the Saudi Press Agency, Maliki said the "control of these types of weapons by terrorist organisations, including Al-Houthi armed militias, represents a threat to regional and global security".
Diab says the Saudi-coalition forces are not too far from the capital, and the Houthis have become widely despised due to shortages of food, water, medicine and money.
For his part, leader of Huthi, Abdul Malik al-Huthi, has described Saleh's attitude as "sedition" and has been prepared for tribal arbitration, after warning that main beneficiary of this confrontation is Saudi coalition.
Abdel-Malek al-Houthi, the leader of the rebels, called on Saleh to "show more wisdom and maturity" in a separate statement.
A statement from Saleh's party called for its loyal armed tribes and supporters to fight against Houthis.
Yemen's war has killed more than 10,000 people since 2015, displaced more than two million people, caused a cholera outbreak infecting almost one million people and led the country on the brink of starvation.
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