The dispute risks escalating to warfare and has caused diplomatic problems beyond their borders. Clinton, shuttling between capitals, told leaders to act quickly toward settling the matter.

“The final steps toward peace often are the most difficult, but we believe peace is possible,” she said at news conference in Baku with her Azeri counterpart, Elmar Mammadyarov. She soon made the one-hour flight to Armenia and drove to the presidential palace in Yerevan for a meeting and dinner with President Serge Sarkisian.

Sarkisian said the dispute over Nogarno-Karabakh was the single most important issue for his country — echoing almost word for word a statement hours earlier by Azeri President Ilham Aliyev.

Azerbaijan, which shares a border with Iran, is of particular strategic importance to the United States. Tens of thousands of flights carrying war supplies to US and allied forces in Afghanistan have crossed Azeri airspace over the past nine years of fighting there since the Sept. 11 terrorist attacks. Azerbaijan also is part of an overland supply chain that is a key alternative to the main land route through Pakistan to Afghanistan.

The US also has good relations with Armenia and has worked to broker an agreement between Armenia and Turkey on establishing formal diplomatic relations and opening their sealed border after a century of enmity. Turkey, however, has refused to ratify the agreement until Armenia removes its military forces of Nagorno-Karabakh, an enclave in Azerbaijan that has been under control of Armenian troops and ethnic Armenian forces since a 1994 cease-fire. The truce ended six years of war that killed about 30,000 people and displaced an estimated 1 million.