Meanwhile, two senior US officials have met Turkish officials in the capital, Ankara, to try to ease tensions after a US congressional resolution branded the mass killings of Armenians by Ottoman Turks after 1915 as genocide.
Some analysts believe the US vote could weaken Washington's influence on fellow Nato member Turkey and increase the likelihood of a Turkish incursion into northern Iraq to attack Kurdish fighters.
Rice said she told the Turkish officials "that we all have an interest in a stable Iraq and that anything that is destabilising is going to be to the detriment of both of our interests".
The secretary of state acknowledged strains following the resolution and said Washington would strive to stop it being approved by the full US congress.
She said: "It is a tough time. It's not an easy time for the relationship and it was perfectly predictable."
On their visit to Ankara, Daniel Fried, US assistant secretary of state and Eric Edelman, undersecretary of defence, faced criticism from Turkish officials over the resolution.
Turkey recalled its ambassador to the US for consultations this week after the vote.
A high-level Turkish diplomatic source who attended their meetings said: "We told him [Edelman] that if the resolution passed in the House it would lead to irreparable damage in our relationship with the United States."
The source, who declined to be named, said Turkish-US co-operation within Nato would also be damaged.
Turkish diplomatic sources said Edelman had promised to remind Iraq of its obligations under the UN charter to protect the borders and fight terrorism.
Edelman also held talks with deputy chief of general staff, General Ergin Saygun, a Turkish diplomat said.
Turkish officials said foreign ministry and military staff met after the resolution was approved to discuss potential measures against the US.
"Why is our government wasting time passing meaningless resolutions? Who cares what America thinks about another country's past?"
Tom Dougherty, Atlanta, USA
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These could include blocking US access to Turkey's Incirlik air base, cancelling procurement contracts, scaling down bilateral visits, denying airspace to US aircraft, and halting joint military exercises, analysts and diplomats say.
Meanwhile, Turkish artillery fired seven to eight shells into a village in northern Iraq late on Saturday, witnesses said.
It was the latest bout of shelling of the mountainous border area where separatist fighters are believed to be hiding.
The witnesses said the shells landed in a village, about 5km from the Turkish border, in Dahuk province.
No casualties or instances of damage were reported.
Ankara has long complained that Washington has not done enough on its own or through the Iraqi government to crack down on the PKK, which uses northern Iraq as a base to attack Turkey.