Parliament has been told that "legitimate" representatives of Nagorno-Karabakh should be allowed to decide the fate of the disputed territory.
Nagorno-Karabakh is an enclave within Azerbaijan, which declared its independence in 1991, and is backed by Armenia. There has been a recent resurgence in the previously frozen conflict with more than 60 people killed and many wounded.
MEPs in Strasbourg debated the current situation with EU foreign affairs chief Federica Mogherini on Tuesday amid calls for the EU to assist in mitigating the crisis.
RELATED CONTENT
Josep-Maria Terricabras, a Vice-Chair of Parliament's Greens/EFA group, said the people of the territory ought to be included in the negotiations about its future.
He explained, "On 10 December 1991, the vast majority of the population of Nagorno-Karabakh voted for independence in a referendum.
"The response to this democratic process was a military offensive by Azerbaijan, with more than 30,000 dead. Subsequently, a ceasefire between Azerbaijan, Nagorno-Karabakh and Armenia was signed.
"The creation of the Minsk Group in 1992 was a breakthrough, but Azerbaijan has refused to implement a true international mechanism for monitoring violations of the ceasefire.
"Nagorno-Karabakh has fully democratic institutions, an exemplary judicial system and low levels of corruption.
"For this reason we request the inclusion of representatives of Nagorno-Karabakh in the negotiations of the Minsk group, precisely because they decide the future of the country's inhabitants.
"Once the military conflict is resolved it will be necessary to agree a procedure to properly recognise the new independent state at international level."
Armenia has been singled out recently for criticism, including from the Strasbourg-based Council of Europe which has voiced "real concern" at a serious increase in hostilities in the so called frozen conflict.
Pedro Agramunt, President of its parliamentary assembly, said he was "extremely worried" at the upsurge in violence.
He issued a statement which called for "the withdrawal of all Armenian armed troops from occupied Azerbaijani territories in compliance with UN Security Council resolutions".
The clashes have been called the "largest-scale hostilities" since a 1994 truce ended a war in which Armenian-backed fighters seized the territory from Azerbaijan.
Agramunt said, "I am extremely worried at the reports of violence in Nagorno-Karabakh and deeply saddened at the loss of life and call on both sides to honour the cease-fire and swiftly resume negotiations towards a peaceful resolution of the Nagorno-Karabakh conflict."
Several UN Security Council resolutions have condemned the use of force against Azerbaijan and the occupation of its territories and reaffirmed its sovereignty and territorial integrity.
The UN resolutions affirm Nagorno-Karabakh's right to be part of Azerbaijan and demand "immediate, complete and unconditional" withdrawal of "occupying forces" from the territories of Azerbaijan.
Azerbaijan has now called on the OSCE to press Armenia to UN honour the Security Council resolutions.
Mogherini, who recently visited Baku, the Azerbaijani capital, in a bid to broker a settlement, urged both parties to stop the fighting, avoid statements that could result in escalation and observe the ceasefire.
The EU recently called the Nagorno-Karabakh status quo "unsustainable".
MEPs recently said that the recent violent clashes in the disputed region underline the need for a complete rethink of the EU's eastern neighbourhood partnership policy.
Both Russia and the West appealed to all sides to show restraint, with key regional power broker Russian President Vladimir Putin calling for an "immediate ceasefire".
The embassy of Azerbaijan to Belgium, Luxembourg and the EU responded to the comments by telling this website, “The Greens/EFA group, as a political group within the European Parliament, has always supported the sovereignty and territorial integrity of Azerbaijan and condemned occupation of our territories by Armenia.
"Several well-known MEPs from the group, such as Ulrike Lunacek, and Heidi Hautala, who is co-chair of the Euronest Parliamentary Assembly, and others have made numerous statements in this regard.
"Of course the EFA is entitled to have its opinion on the issue but what matters is the official position of the Greens/EFA political group and, essentially, the official position of the European Parliament which unequivocally supports the sovereignty and territorial integrity of Azerbaijan and has deplored occupation of its territories by Armenia in its various resolutions."