US Secretary of State Pompeo aims to press on with mediation efforts to end the conflict in the region.
Hopes of ending nearly a month of bloodshed in the occupied-Karabakh have looked slim as Azerbaijani and Armenian forces fought new battles on the eve of talks in Washington.
Plans for US Secretary of State Mike Pompeo to meet the foreign ministers of Azerbaijan and Armenia on Friday raised hopes this week that the two former Soviet republics would agree to end their deadliest fighting since the mid-1990s.
Russian President Vladimir Putin said he hoped that the United States would help Moscow broker a solution to 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.
He added that he speaks to leaders of Armenia and Azerbaijan several times a day by phone.
But those hopes have been dented by the continued heavy fighting in and around Nagorno-Karabakh, a breakaway territory that is inside Azerbaijan but controlled by ethnic Armenians, and by angry rhetoric from both sides.
Heavy losses from both sides
Hundreds of people have been killed since fighting flared on September 27, raising fears of a wider war drawing in Turkey and Russia and increasing concern about the security of pipelines in Azerbaijan that carry Azerbaijani gas and oil to world markets.
Putin said on Thursday that Moscow believed that nearly 5,000 people had been killed in fighting between Azerbaijan and Armenian forces, with more than 2,000 dead on each side.
Occupied-Karabakh said 874 of its military personnel had been killed since September 27, in addition to 37 civilians.
Azerbaijan says 63 Azeri civilians have been killed and 293 wounded, but has not disclosed its military casualties.
Armenian Prime Minister Nikol Pashinyan said on Wednesday that he could see no diplomatic resolution for the long-running conflict at this stage.
In a transcript of comments to the Nikkei newspaper published on Thursday, Azerbaijani President Ilham Aliyev said the prospects of reaching a peace settlement were "very remote".
Reiterating Azerbaijan's main condition for ending the fighting, Aliyev demanded promises that his country will be handed back control of occupied-Karabakh, which broke away as the Soviet Union collapsed.
"So our main objective at these discussions will be to find out whether the Armenian leadership is ready to liberate our territories or not, and if ready, then when?" he said.
Armenians regard occupied-Karabakh as part of their historic homeland and accuse Azerbaijan of making a land grab in the recent fighting.
But the entire region is internationally-recognised as being part of Azerbaijan.
Aliyev said he would not rule out "cultural autonomy" for ethnic Armenians in the region, but not say what he meant by this.
Pompeo said he still hoped a diplomatic solution could be found as the US, France and Russia press on with mediation efforts they have led for decades.
He said the "right path forward is to cease the conflict, tell them to de-escalate, that every country should stay out".