According to Azad Jundiyani, a member of Iraqi President Jalal Talabani's Patriotic Union of Kurdistan in Suleymaniyah, Barzani issued a formal message asking for the Iraqi flag to be lowered.
The message was also broadcast on Kurdish radio on Thursday. The Kurdish flag is already flown outside government buildings throughout Kurdistan.
A move for Kurdish autonomy has slowly been gathering momentum since the 2003 US invasion of Iraq.
Sunni Arabs fear that Kurds are pushing for secession under the nation's new federal system, a step which, if imitated by the Shia majority in the oil-rich south, would leave Sunnis little national resources.
In May, the Kurdish parliament in the northern city of Arbil unified the Kurdish region's two long-standing administrations, one headed by Barzani's Kurdistan Democratic Party and the other by Talabani's Patriotic Union of Kurdistan.
Meanwhile, the US Defence Department admitted on Friday that sectarian violence is spreading in Iraq, and is reflecting the most complex security challenges since the invasion in 2003.
The Pentagon said illegal militias have become more entrenched, especially in Baghdad neighbourhoods where they are seen as providers of security as well as basic social services.
The report described a rising tide of sectarian violence, fed in part by interference from neighbouring Iran and Syria and driven by a "vocal minority" of religious extremists who oppose the idea of a democratic Iraq.
Iraqi police are a favoured target
of anti-government fighters
Death squads targeting mainly Iraqi civilians are a growing problem, heightening the risk of civil war, the report said.
"Death squads and terrorists are locked in mutually reinforcing cycles of sectarian strife," the report said.
It added that the Sunni-led armed anti-government campaign "remains potent and viable" even as it is overshadowed by the sect-on-sect killing.
"Conditions that could lead to civil war exist in Iraq, specifically in and around Baghdad, and concern about civil war within the Iraqi civilian population has increased in recent months," the report said.
It is the latest in a series of quarterly reports required by Congress to assess economic, political and security progress in Iraq.
Growing numbers of Congress members are recommending either a shift in the Bush administration's Iraq strategy or a timetable for beginning a substantial withdrawal of American forces.
According to Iraq's interior ministry violent deaths among civilians in Iraq may have fallen by a quarter in August compared with July, despite a bloody week in Baghdad that ended with 70 dead in a series of explosions is said on Friday.
The partial data, provided by the interior ministry and based on figures from the health ministry, tend to confirm US military confidence that a crackdown in the capital has slowed the bloodletting but also that dozens are still dying every day.
A Pentagon report showed Iraqi deaths in the past quarter rose by a half over three months earlier, as sectarian strife has become the "core conflict" and created a risk of civil war.
The Pentagon's quarterly report on Iraq conceded that, in the most complex situation since the 2003 invasion ousted Saddam Hussein, "conditions exist that could lead to civil war".
"Nevertheless, the current violence is not a civil war, and movement towards a civil war can be prevented."