The coalition has also been accused of committing war crimes through indiscriminate attacks on residential areas in cities controlled by the rebel forces.
Saleh stepped down after 33 years in office in 2012, following months of Arab Spring protests against his rule, but remained leader of the General People's Congress Party (GPC), the country's largest political party.
The clashes between Mr Saleh's supporters and the Houthis underscore the complex situation in Yemen, one of the poorest countries in the Middle East, where a proxy war between the Iran-aligned Houthis and the Saudi-backed Mr Hadi has caused one of the worst humanitarian catastrophes in recent times.
Citing the restricted access to his country, Saleh said Saturday in a televised address on Yemen Today TV that he is open to talks with the Saudi-led coalition.
Since March 2015, Saudi Arabia and some of its Arab allies have been carrying out deadly airstrikes against the Houthi Ansarullah movement in an attempt to restore power to fugitive former president Abd Rabbuh Mansour Hadi, a close ally of Riyadh.
"We vow to our brothers and neighbors that, after a ceasefire is in place and the blockade is lifted. we will hold dialogue directly through the legitimate authority represented by our parliament".
"The General People's Congress and its allies hold Ansar Allah fully responsible for every drop of blood shed among the Yemenis. and warn against all acts that, rather than serve national unity, threaten our internal unity and cohesion", the party said in a statement.
"We will deal with them in a positive way and what happened to Yemen is enough", he added.
The Saudi-led coalition welcomed Mr Saleh's change of stance.
Officials loyal to Saleh began to report that their supporters and fighters had seized control of the state TV stations, airport, camps in the capital.
It also called on supporters, including tribal fighters, to "defend themselves, their country, their revolution and their republic".
The head of the Houthis' Ansarullah group warned that the biggest victor from what he described as Saleh's "sedition" was the Saudi-led coalition. He fought six wars against the Huthis when he was president, but joined forces with them to take over the capital in 2014.
The Houthi group leader, Abdel-Malek al-Houthi, had earlier appealed to Mr Saleh to avoid any escalation, saying that the crisis would only serve Yemen's enemies.