Robert Gates, the US defence secretary, on Sunday called for a short ground campaign and urged Turkey to do more to reconcile with its Kurdish minority.
He said: "In terms of the current operations, I would hope that it would be short, that it would be precise and avoid the loss of innocent life and that they leave as quickly as they can accomplish the mission."
In its statement, the Turkish General Staff confirmed the deaths of just seven of its soldiers, two of them on Saturday.
It is virtually impossible to verify the claims of either side because the fighting is taking place in largely inaccessible terrain in tough winter conditions.
There have been conflicting reports about the scale of the cross-border operation, the largest in a decade.
The General Staff has not said how many troops are involved, but it said on Saturday fighting was raging in four different areas of northern Iraq, suggesting a large-scale operation.
A senior Turkish military source said two brigades made up of around 8,000 troops were taking part, while Turkish media have put the number of troops at 10,000.
But a senior officer with US-led forces in Baghdad said the number was under 1,000.
The Turkish offensive has concerned the US and EU, which fear a prolonged military campaign inside Iraq would raise the risk of serious clashes between Turkish and Iraqi Kurdish forces and could undermine the fragile US-backed government in Baghdad.
However, international reaction has been relatively muted.
Some analysts argue this is due to regular air assaults by Turkey and greater stability within Iraq.
Laurence Lee, Al Jazeera's correspondent in Istanbul, said Turkey had seen "a window of opportunity" and launched it offensive.
"But it is extraordinary that the only people that have come out against this is the Kurdish administration in northern Iraq and that everybody else seems to have suddenly accepted that it is okay to do,' Lee said.
Ali al-Dabbagh, the Iraqi government's spokesman, acknowledged that Turkey was "suffering from the terrorist PKK organisation" but that military action was not the answer.
"We know the threats that Turkey is facing but military operations will not solve the PKK problem," he said.
He said Turkish commanders had assured Iraq that the "operation will be a limited one and it will not violate certain standards that they have set".
A spokesman for Nuri al-Maliki, the Iraqi prime minister, said on Thursday that he had telephoned Reccip Tayyib Erdogan, his Turkish counterpart, urging him to respect Iraq's sovereignty.
'Martyrs do not die'
Ali Babacan, the Turkish foreign minister, reiterated on Saturday that the PKK was the sole target of the northern Iraq offensive.
The Turkish General Staff said: "The operation will end once our targets have been reached."
The Turkish military posted 11 photographs of the cross-border operation on its website. In one, 11 troops in white camouflage hold machine guns on a snow-covered hill.
A Turkish Air Force pilot is seen inside his aircraft in another picture. Another shows a Turkish soldier writing "Martyrs do not die" on two unused artillery shells.
The phrase is often repeated by crowds at the funerals of soldiers killed in fighting.
The ground offensive started after Turkish fighter jets and artillery bombed suspected PKK targets on Thursday, the Turkish military said on its website.