He says a new page must be opened in relations (with Saudi Arabia) and neighborly ties restored, putting an end to all (the animosity) that has taken place and that amicable relations can resume once there is a cease-fire (with the Saudi-led coalition) and their blockade of Yemeni ports and airports is lifted.
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.
"Such behaviors are unjustifiable", al-Houthi said on Saturday in response to measures taken by supporters of Saleh, the head of Yemeni General People's Congress (GPC) party, against the Houthi Ansarullah movement.
In March 2015, warplanes from Saudi Arabia, the UAE, Kuwait, Bahrain, Qatar and other allies began airstrikes on rebel positions in Yemen.
The source said that clashes had renewed on Saturday between the Houthis and Saleh's supporters in the Yemeni capital Sana'a. 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.
Street fighting in the war-wracked country comes amid heightened regional tensions, most notably after Houthi rebels early last month launched a missile that was intercepted near the Saudi capital. The country has since fallen into chaos and Mr Saleh later joined the Houthis to drive Mr Hadi out of the capital in 2014.
A Houthi spokesman was quick to denounce Saleh's comments, accusing the former president of staging a "coup".
Saleh called for a joint ceasefire to be agreed on by his loyalists and the Houthi militia, after clashes entered their fourth day and hae resulted in the deaths of at least 80 people.
The United Nations has warned Yemen faces mass starvation unless the Saudi-led coalition allows more food aid to enter the impoverished country.
Meanwhile, Houthi leader, Abdel-Malek al-Houthi, called for dialogue with Saleh and his rebel forces to end the growing violence between the two factions.
A statement from Saleh's party called for its loyal armed tribes and supporters to fight against Houthis.
The conflict and a blockade by the coalition has also left more than 20 million people in need of humanitarian aid, created the world's largest food security emergency, and led to a cholera outbreak that is thought to have killed 2,211 people since April.