The coalition, which has fought Saleh and the Iran-backed Houthis since 2015, last month imposed a total blockade on the impoverished country after a rebel missile was shot down near Riyadh.
But Saleh's message didn't sit well with the Houthis, whose spokesman called his remarks "a coup against our alliance and partnership".
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.
"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 Qatari regime had been attempting since Friday to mediate a rapprochement between the Houthi militias and former Yemeni President Ali Abdullah Saleh's forces, an informed source told Asharq Al-Awsat on Saturday.
"We will deal with them in a positive way and what happened to Yemen is enough", he added.
Mr Saleh, who led Yemen for more than 30 years, was deposed in a 2011 uprising.
The Saudi-led coalition, consisting of several Arab-Sunni states, intervened in neighbouring Yemen to push back Houthi rebels and has since been trying to reinstate the government of President Abd-Rabbu Mansour Hadi.
Residents of Sanaa described heavy fighting on the streets of Hadda, a southern residential district of the Yemeni capital where many of Saleh's relatives, including his nephew Tareq, live, early on Saturday, with sounds of explosions and gunfire heard all over the area. The Houthi allied themselves with the forces loyal to Saleh and have fought a devastating war against forces loyal to Hadi and his foreign backers.
There was no immediate word on casualties.
"The GPC appealed to the army and security forces to remain neutral in the conflict".
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.
Stressing that Yemen's interests lie in avoiding sedition, the Houthi leader called on citizens and tribal fighters to focus on the battle against the invading Saudi regime and stay away from any provocative acts.
For his part, Houthi leader Abdel Malek al-Houthi called on Saleh to end what he called "the sectarian strife" that he has instigated, and threatened to take action against his supporters militarily if he does not.