The popular initiative by Thirdway Alliance party could suffer in county assemblies following President Uhuru Kenyatta's party decision to reject it.
Even after garnering over 1 million signatures, Dr Ekuru Aukot still needs backing from majority of county assemblies, a move that is now now granted.
Jubilee party on Tuesday joined a team of major parties which have already rejected the bill, terming it as a shortcut to inspire constitutional changes.
Raila Odinga's ODM, ANC of Musalia Mudavadi, Wiper of Kalonzo Musyoka and Ford Kenya of Moses Wetangula have all rejected the bill.
Secretary General Raphael Tuju on Tuesday said Jubilee will only support a referendum that will come about after recommendations by BBI team.
“The BBI is an initiative of our party leader — President Uhuru Kenyatta — together with former Prime Minister Raila Odinga and is even funded by the government.
“It’s a little bit stretching for us to have an initiative initiated by the government and our party leader and to be preoccupied in any way with a parallel initiative,” Mr Tuju said.
The Jubilee Party official pointed out that BBI was an inclusive initiative “which has got some elder statesmen going around the country getting views and I think they have now finished getting opinions from Kenyans on how they want to be governed.”
He said Punguza Mizigo initiative did not involve jointly the people of Kenya, thus making it unpopular among the Jubilee party members.
“Now, however, brilliant we may be, a Constitution is about a contract and a contract should involve discussion between the two — the governed and the government,” the Jubilee Secretary General said.