White House says withdrawal to be based on “conditions on the ground”.
The US military says that the northern city is the last urban area in Iraq where al-Qaeda remains strong.
It said that “eight Iraqi civilians killed, 33 civilians were wounded and four Iraqi soldiers were wounded” in what it said was a dual car bomb attack against the general.
The US and Iraqi forces are engaged in a military crackdown in Mosul that began on May 14.
Elsewhere in Iraq, six other people were killed in two near-simultaneous bombings, according to security officials.
The bombs exploded at around 6:30am (03:30 GMT) near a bank in the centre of Falluja in the western Anbar province.
The dead included four policemen, a security official said.
The second bomb went off as officers were aiding victims of the first attack, he said.
The Iraqi military said on Wednesday that “terrorist attacks” in June were down 85 per cent on the same period in 2007.
An average of 25 attacks took place each day in June, compared with 160 during the same month last year, Major General Qassim al-Mousawi, an Iraqi army spokesman, said.