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.
Hopes of ending almost a month of bloodshed in the mountain enclave of Nagorno-Karabakh looked slim on Thursday as Azeri and ethnic Armenian forces fought new battles on the eve of talks in Washington.
But the entire region is internationally-recognised as being part of Azerbaijan.
According to Nagorno-Karabakh officials, 900 of their troops have been killed, and more than 30 civilians.
Azerbaijan says 61 Azeri civilians have been killed and 291 wounded, but has not disclosed its military casualties.
Russian President Vladimir Putin said Thursday that Moscow's information indicated that the combined death toll from the almost four weeks of fighting was approaching 5,000, a significantly higher number than officially reported by the warring parties.
The Trump administration waded cautiously into global efforts to halt fighting between Armenia and Azerbaijan, which are engaged in their worst conflict in more than 25 years.
Armenia's prime minister said on Wednesday he saw no possibility of a diplomatic solution at this stage in the conflict with Azerbaijan over the mountain enclave of Nagorno-Karabakh.
"There is no way now to settle the Nagorno-Karabakh issue through diplomacy", Pashinian said, emphasizing that "there is no Armenia without Nagorno-Karabakh".
The Organization for Security and Co-operation in Europe (OSCE) Minsk Group - co-chaired by France, Russia, and the U.S. - was formed in 1992 to find a peaceful solution to the conflict, but to no avail.
However, it is widely understood that the death tolls are higher than what has been reported.
"For us, Armenia and Azerbaijan are equal partners, and it's a huge tragedy for us to see people die", Putin said.
Differences over the conflict have further strained relations between Ankara and its North Atlantic Treaty Organisation allies.
Armenians regard occupied-Karabakh as part of their historic homeland and accuse Azerbaijan of making a land grab in the recent fighting.
Turkey has said it would send soldiers and provide military support for Azerbaijan if such a request were made by its ally.
"Azerbaijan and Turkey have agreements on military co-operation", he said on CNN Turk".
Putin said Russian Federation disagreed with Turkey on Nagorno-Karabakh, but both countries needed to find a compromise.
Asked about Zatulin's statement, Putin's spokesman, Dmitry Peskov, responded that "there is no alternative to a peaceful settlement".
Speaking at the end of an interview with the Armenian president on the sidelines of NATO defense ministers' virtual meeting, NATO Secretary-General Jens Stoltenberg assured that his organization supports a strict ceasefire application and highlighted its good relations with both parties in the conflict.
But Azerbaijan says mediation has achieved nothing in three decades and wants Turkey involved in peacemaking.