The Building Bridges Initiative (BBI) secretariat has so far collected 1.5 million signatures.
The team on Monday said it is expecting to collect at least 2 million signatures by the end of the day.
The signatures collected so far include those appended on the online app that was launched last week by President Uhuru Kenyatta and Raila Odinga at KICC.
“This is a civic duty, not political event. No citizen should be left behind,” secretariat co-chair Junet Mohamed said.
The exercise is expected to end on Thursday this week.
“Latest, by Thursday we should be done,” Junet added.
In a statement, the team urged those against the Bill to read the document which they maintain will benefit future generations.
They insisted that the document is not about the president or the ODM party leader but about the “prosperity and posterity of our great country”.
BBI opponents have in recent days castigated a move by the government to have chiefs spearhead the drive.
Civil society groups and some politicians allied to Deputy President William Ruto have said that the use of civil servants is a way of turning a political process into a state project.
Signatures and the Constitution (Amendment) Bill 2020 are expected to be submitted to the Independent Electoral and Boundaries Commission on December 9 for verification.
Should the verification process be successful, the draft Bill will be submitted to each county assembly for consideration within three months after the date it was submitted.