Firefighters are working to extinguish the flames at the site, which produces approximately 25,000 barrels of oil per day, oil and security sources told the news outlet.
Iraqi outlet Sumaria News also reported the bombings.
It is currently unclear who is responsible for the incident.
Security has been a constant concern in disputed areas like Kirkuk. The Islamic State
...formerly ISIS or ISIL, depending on your preference. Before that they were al-Qaeda in Iraq, as shaped by Abu Musab Zarqawi. They're really very devout, committing every atrocity they can find in the Koran and inventing a few more. They fling Allaharound with every other sentence, but to hear western pols talk they're not
reallyMoslems....
(ISIS), despite being territorially defeated in Iraq in December 2017, has exploited security gaps between Iraqi forces and Kurdish Peshmerga in the province, with remnants carrying out numerous attacks against security forces and civilians.
Two coppers guarding the Khabbaz oilfield were killed by ISIS
On Thursday the group's weekly propaganda newspaper al-Naba claimed the group has killed and injured at least 19 security forces in Kirkuk in the past two weeks.
An explosion in Kirkuk last month killed two members of Iraqi security forces and injured six others. The attack was later claimed by ISIS.
Members of Kirkuk’s Majlis al-Arabi (Arab Council) and Patriotic Union of Kurdistan (PUK) met with Kirkuk Governor Rakan al-Jabouri in Baghdad’s Salam Palace last week to discuss increasing security through a "united force." The meeting was led by Iraqi President Barham Salih.
"Arabs, Kurds, and Turkmen all agree that there should be a united force defending Kirkuk consisting of the people of Kirkuk themselves, and we have discussed this idea in our meeting," Hatam al-Tai, a spokesperson for the Arab Council told Rudaw English on Monday.
In a statement issued through its official channel on Telegram, Islamic State claimed responsibility for the attack, though it did not provide any evidence to back up its claim.
Technical teams isolated the two burning oil wells and there was no impact on output, two sources from the state-run North Oil Company (NOC), who spoke on condition of anonymity, said.
The field produces about 25,000 barrels per day, oil officials said. The ministry said production from the two wells that were targeted did not exceed 2,000 bpd.
The ministry statement said a fire erupted at the two oil wells after explosive devices were set off half an hour apart, with one going off at 1:30 a.m. (1030 GMT) and the second at 2:00 a.m (1100 GMT).