The four Nasa principals Monday closed ranks to set the ground for constitutional change around the Bomas Draft.
They would like the country to not only have an elected president and a deputy, but also an appointed prime minister, who will head the Cabinet.
However, the draft that the National Super Alliance (Nasa) is now rooting for could be a long way coming as it was altered drastically after the Bomas talks in 2005, leading to the rejection of the final document in a national referendum opposed by Mr Raila Odinga, Mr Kalonzo Musyoka, Mr Musalia Mudavadi, Mr Uhuru Kenyatta and Mr William Ruto, now the President and Deputy President, respectively.
The independence Constitution was changed in a 2010 referendum -- ushering in a purely presidential system that had been opposed by Mr Odinga's camp before they made a U-turn and backed it.
But, following the protracted, high-stakes elections in 2013 and 2017, Nasa now argues that the time is ripe for Kenyans to reconsider the document, which the opposition coalition's principals said Monday is "what Kenyans wanted."
Mr Odinga, Mr Mudavadi and Mr Moses Wetang'ula agreed at a meeting in Nairobi to revive debate on the draft, which they believe is the best guide to end regular political conflicts blamed on the winner-take-all system.
"The Bomas Draft contains cures to the problems of inclusion, strengthening of devolution, the shape of the Executive, and security sector reforms," Nasa leaders had said in a statement issued on Friday after a meeting at Mr Odinga's Capitol Hill office.
Monday's meeting, which started at 4pm, came amid renewed debate on changing the Constitution.
Mr Odinga and Ford-Kenya leader Wetang'ula were both quoted at the weekend as calling for talks on the overhaul of the seven-and-a-half-year-old document.
"Events in the country are crying out for a non-contested constitutional referendum," Kitui Senator Enoch Wambua, a close ally of Mr Musyoka, said of the renewed Nasa push.
"The Bomas Draft is perhaps the best reference document in the said review. If Jubilee and what is left of Nasa cannot agree on anything else, at least we must be able to agree on the need to fix what is not working for Kenya in the Constitution."
Produced at the Bomas of Kenya Delegates' Conference in 2005 -- spearheaded by law professor Yash Pal Ghai -- the document that came to be known as the Bomas Draft proposed that Kenya has a President, a Deputy President, a Prime Minister, and several ministers.
The Prime Minister, who had to be the leader of the largest political party or coalition in the National Assembly, would be appointed by the President from among MPs with the approval of Parliament.
The Premier would be the head of a Cabinet comprising two deputy premiers, a maximum of 20 and a minimum of 15 ministers, and an equal number of deputy ministers.
The ministers, the document proposed, would be appointed by the President upon nomination by the Prime Minister from among members of the National Assembly, and subject to the approval of the Senate.
"A hybrid parliamentary system based on the Bomas Draft will not only allow individuals from smaller communities to rise to the highest office in the land, but also encourage an inclusive system of power sharing," Nasa states in its 44-page August 2017 elections manifesto.
While the Premier would head Cabinet and chair its meetings, the Bomas document allows the President to retain powers to hire and fire the Cabinet, judges, as well as ambassadors.
In the recommendations that Nasa now wants examined afresh, the Deputy President will remain the President's principal assistant, with a full mandate to take over if the Head of State is incapacitated.
Orange Democratic Movement secretary-general Edwin Sifuna, while confirming Mr Odinga's meeting with his allies Monday said the Bomas Draft was becoming attractive because "it was a people-driven process".
"We will always hold the Bomas Draft as the best expression of the people's aspirations, but right now the focus is on electoral justice," said Mr Sifuna.
It was not clear whether Nasa's new push will be executed through the proposed people's assemblies and the national convention.
In its statement after the Friday meeting last week, the coalition said the adoption and implementation of the Bomas Draft "will have to be consultative and people-driven and not an exclusive affair negotiated in Parliament, hence the need for a national dialogue on the country's future".