Raila Odinga's Orange Democratic Movement (ODM) now wants a referendum to be held in April, saying all signs are that issues that could have derailed the process have been ironed out.
Mr Opiyo Wandayi, the party's Director of Political Affairs, on Sunday asked the BBI Secretariat to fast-track the process and fine-tune the Round Two recommendations so that a referendum is held by April.
Speaking in his constituency when he issued driving licences to 276 youths from Ugunja, the MP said that the events that transpired during the BBI consultative meeting in Mombasa on Saturday were a clear sign that the country had agreed on the contentious issues that would have delayed the process.
He recommended that the Yusuf Haji-led team should immediately start drafting bills on issues that will require legislation before April, saying that the taskforce has no reason to wait until June as stipulated in their extended term.
"A lot of ground has been covered and issues that were hitherto contentious have now been agreed upon, including suggestions to create the office of a powerful Prime Minister and two Deputies. Both sides of the political divide on Saturday demonstrated support for BBI in totality," he said.
The MP said the issue of a powerful prime minister which had been opposed by a section of politicians, especially those allied to Deputy President William Ruto, had been dealt with after his troops, led by Elgeyo Marakwet Senator Kipchumba Murkomen, indicated that they had no problem with the proposal.
Mr Wandayi reiterated ODM's position on the issue while insisting that the country needs an executive PM to check the president.
"We don't want power concentrated in the presidency. Power must be dispersed. A powerful prime minister will check the excesses of an imperial president," he said, adding that the party was opposed to the creation of a toothless PM.
Mr Wandayi, who chairs the Parliamentary Accounts Committee (PAC), said the party did not oppose the proposal to create regional governments but told off a section of governors who, he said, must stop thinking that the position was being created for them.
"The regional positions are being proposed because they are necessary for strengthening devolution and will be up for grabs by all potential candidates in the country," he said.
Mr Wandayi asked second-term governors who are already positioning themselves to get the slots to be ready to battle it out with other contestants.
"There is no guarantee that the second-term governors will automatically occupy the regional seats," he said.
The legislator further said that the country had enough resources to fund the offices that are being suggested in the constitutional changes, adding that the biggest problem the country must tackle is corruption.
He said that the National Assembly will be keenly monitoring how the Senate will handle the impeachment report of Kiambu Governor Ferdinand Waititu.
"We ask all ODM senators to stand with the people of Kiambu. Those who will support the besieged governor will be supporting corruption," he said.
He also asked the ODM legislators to speak out against corruption both at the national government and in the counties.
"We must at all times fight corruption if we want to end the problems bedevilling the country especially in the county governments," he said.
After the conclusion of the BBI process, he said, ODM will hold its grassroots polls.
"We will give you an opportunity to elect new party officials and weed out those who have not performed well through a democratic process."