The central Najaf and Karbala provinces could be under full control by Iraqi police within three months, while Babil and Baghdad to the north of them could be ready for such a move by December, said the official on Saturday.
Since his national unity government was sworn in a week ago after months of sectarian wrangling, Nuri al-Maliki, the Iraqi prime minister, has talked up the speed at which his newly trained forces can replace Americans.
He said they could be in control of most of Iraq by the end of 2006 and all of it next year.
Security analysts are sceptical and many expect some foreign troops to be in Iraq for years to come.
In other news, the US Marine Corps said an AH-1 Cobra helicopter went down in Iraq's Anbar province and two crew members are missing. Hostile fire is not being suspected as the cause of the crash, the military said.
The helicopter was on a maintenance test flight on Saturday when it went down and search and rescue efforts are ongoing for the missing crew members, the statement added.
"We are using all the resources available to find our missing comrades," said Lieutenant-Colonel Bryan Salas, a marine spokesman.
The statement added that "the incident does not appear to be a result of enemy action".Mottaki visit
Meanwhile, a visit to Iraq by Manouchehr Mottaki, the Iranian foreign minister, prompted grumbling by Sunni politicians about influence over Iraq from the Islamic Republic, where many of Iraq's leading Shia politicians found refuge from Saddam.
Mottaki (L) met a range of clerics
in Karbala and Najaf
On Saturday, Mottaki met a range of clerics in Karbala and Najaf, including Iraq's senior religious figure Grand Ayatollah Ali al-Sistani, whom he praised for working for Iraqi unity. He also visited holy sites for Shias.
Mottaki on Friday ruled out talks for the time being with US officials, political sources said. Washington, which says Iran is fomenting violence in Iraq, had said it was open to discussions with its old adversary.
There was no immediate reaction from Washington to Mottaki's rejection of talks.