Official results for the election held on the same day as Iraq's national vote have not yet been released.
But Turkmen and Arab parties in the northern city on Sunday said that Kurds from other parts of the country flooded the city on election day to inflate the community's vote.
According to local Kurdish media, the Kurdish list expects to get about 63% of the vote for the Tamim provincial council which includes Kirkuk.
"The elections lack credibility because of the major violations and the absence of international observers," a Turkmen candidate for the provincial election, Saad al-Din Arkaj, said after a meeting of Turkmen parties in Kirkuk.
Arkaj said the Iraqi election commission should review the whole vote count and investigate the complaints of Arab and Turkmen parties.
"The results fixed by the Kurds will cause a catastrophe," he warned.
Kurds deny seeking to wipe out
Turkmen and Arab presence
"The Turkmen cannot accept this plot through which the Kurds want to join Kirkuk to Kurdistan."
An Arab candidate for the provincial council also called the election organisation a "plot".
Abd al-Rahman Munshid al-Assi said thousands of Kurds had been brought to the city from Sulaimaniya and Arbil provinces on 30 January "to vote a second time in Kirkuk".
"They are aiming to attach Kirkuk to Kurdistan and we Arabs and Turkmen reject this," he said after a meeting of Arab and Turkmen parties on Saturday night.
"We are examining all options as we will not have a real presence on the provincial council. Two thirds of the seats will go to Kurds," al-Assi predicted. Kurdish leaders deny any vote fraud.
"Unfortunately, the Arabs and Turkmen do not understand democracy," Rajkar Ali, a candidate for the Patriotic Union of Kurdistan (PUK) for the election, said.
"We are not seeking to wipe out the presence of our Arab and Turkmen friends. We want to win back the rights that were taken from the Kurds," he said.
"They are aiming to attach Kirkuk to Kurdistan and we Arabs and Turkmen reject this"
Abd al-Rahman Munshid al-Assi,
Arab provincial candidate
A decision brokered in January by the US selected interim Iraqi government gave tens of thousands of displaced Kurds the right to vote in Kirkuk, effectively tipping the balance to the Kurdish community and drawing the ire of neighbouring Turkey.
Sunni and Shia Arab parties withdrew from the local election in Tamim province in protest.
Former Iraqi president Saddam Hussein is said to have sought to Arabise Kirkuk by moving Kurds away and bringing in Arabs from other parts of the country.
Leaders of the Kurdish autonomous region want Kirkuk as part of their region.