Although the U.S. has officially voiced neutrality over the disputed region, Pompeo recently characterized Armenia's actions as defensive and has criticized the involvement of Turkey, a staunch ally of Azerbaijan, who - along with Israel - has helped to arm the Muslim-majority state.
On October 23, in the framework of the working visit to the United States, Foreign Minister of Armenia Zohrab Mnatsaknayan held an online meeting with Nancy Pelosi, the Speaker of the United States House of Representatives.
But hours before talks were due to start in Washington, DC, Azerbaijani and ethnic Armenian forces clashed again in the fight over the breakaway Nagorno-Karabakh region.
Shortly before the Washington talks, Turkish President Tayyip Erdogan told reporters in Istanbul that he hoped Moscow and Ankara could work together on resolving the conflict.
"I very much hope that our American partners will act in unison with us and will help the settlement", Putin told a meeting of the Valdai Discussion Club.
His statement came after several rockets fired by Azerbaijani and Armenian forces fell on Iranian villages and farms in the province.
Fighting between Armenia and Azerbaijan over Nagorno-Karabakh, also called Upper Karabakh, began September 27.
Hundreds of people have been killed since fighting flared on September 27, raising fears of a wider war drawing in Turkey and Russian Federation and increasing concern about the security of pipelines in Azerbaijan that carry Azeri gas and oil to world markets.
Putin said on Thursday that Moscow believed that almost 5,000 people had been killed in fighting between Azerbaijan and Armenian forces, with more than 2,000 dead on each side. Azerbaijan does not disclose its military death toll.
President Ilham Aliyev told French newspaper Le Figaro that Azerbaijan was ready to sit down for negotiations but blamed Armenia's actions for the continued hostilities.
Armenian Prime Minister Nikol Pashinyan said on Wednesday he could see no diplomatic resolution of the long-running conflict at this stage.
Azerbaijan's defence ministry reported fighting in and around Nagorno-Karabakh, a part of Azerbaijan populated and controlled by ethnic Armenians.
But the entire region is internationally-recognised as being part of Azerbaijan. Nagorno-Karabakh's ethnic Armenian administration says its forces have repulsed attacks.
"We are committed to finding a political solution [to] the conflict and ready to resume substantive talks immediately", he said in a statement after the talks.
Pompeo said he hoped a diplomatic solution and the "right path forward" could be found as the United States, France and Russian Federation press on with the mediation efforts they have led for decades. Pompeo has called on other countries not to provide "fuel" for the conflict.