The draft bill was adopted by Iraq's 275-member parliament on July 22, but only 140 MPs were in the chamber to vote as many Kurds and some Shia ministers chose to stage a boycott.
The vote forced the three-member presidency council, headed by Jalal Talabani, the Kurdish Iraqi president, to send the legislation back to the parliament for reworking.
Article 24 of the draft bill seeks to equally divide power in the regional council among Kirkuk's three main groups, Arabs, Kurds and Turkmen. But Kurds, who constitute a majority are vigorously opposed to the plan.
"Kurds are for the distribution of authority in Kirkuk but not equally dividing it," Barzani said.
"The separation should come from the election results."
Since the bill came before the house in June, Kurds in Arbil have staged a series of angry demonstrations, while Arabs have organised their own protests in a Kirkuk suburb and Baghdad, Iraq's capital.
In the latest rally, several hundred people in a Sunni neighbourhood of Baghdad took to the streets on Monday to protest what they see as moves by Kurds to incorporate Kirkuk into the autonomous Kurdish region.
Meanwhile, a large number of Iraqi parliament members are stalling on a final date for the polls, with the country's election commission saying the date was unrealistic.
Washington has been pushing for the elections for more than a year as its views the provincial poll as a key step to healing the sectarian divides that have torn the nation apart since the US-led invasion in 2003.