US Defence Secretary Robert Gates has urged Turkey to keep its military campaign against Kurdish militants in northern Iraq as short as possible.
Iraq has said the Turkish incursion could destabilise the region
"The shorter the better," he said, adding that Ankara should attempt to reconcile with its Kurdish minority.
Turkish troops began an offensive against the Kurdistan Workers' Party, the PKK, on Thursday.
The PKK says it has shot down a Turkish Cobra attack helicopter during clashes in the area.
Ahmed Danees, head of foreign relations for the PKK, made the claim - which has not been independently verified.
"At 6pm (1500 GMT) yesterday, our fighters shot down a Cobra helicopter," he told Reuters news agency.
He added that this had happened in the remote Chamsku area, close to the border.
Speaking at the end of a visit to Australia, Mr Gates said military strikes by Turkey would not be enough on their own to end the long-running dispute with Kurdish rebels.
He said Ankara should employ political and economic measures to isolate the PKK and erode its support base.
Gates urged Turkey to respect Iraq's sovereignty
He said America's experiences in Iraq and Afghanistan showed that military muscle should be complemented by efforts to address grievances held by minority groups.
"These economic and political measures are really important because after a certain point people become inured to military attacks," he said.
"And if you don't blend them with these kinds of non-military initiatives then at a certain point the military efforts become less and less effective."
Mr Gates also urged Turkey to respect Iraq's sovereignty.
The American defence secretary will deliver his advice in person when he travels to Ankara next week.
The Turkish authorities launched the cross-border attack after accusing the Iraqi government of failing to stop members of the PKK from using the area as a safe haven.
Iraq's foreign minister has warned that any escalation of Turkey's operation against Kurdish rebels in northern Iraq could destabilise the region.
More than 30,000 people have been killed since the PKK began fighting for a Kurdish homeland in south-eastern Turkey in 1984.
The US, the EU and Turkey consider the PKK to be a terrorist organisation.