MEPs say that the recent violent clashes in the disputed region of Nagorno-Karabakh underlines the need for a complete rethink of the EU's eastern neighbourhood partnership policy (ENP).
Nagorno-Karabakh has been in the hands of ethnic Armenian separatists since a war that ended in 1994.
But recent fighting between Armenian and Azerbaijaini forces have left 30 soldiers dead and caused civilian casualties.
The so-called "frozen conflict" between the two South Caucasus countries began in 1988, when Armenia made territorial claims against Azerbaijan. As a result of the ensuing war, in 1992 Armenian armed forces occupied 20 per cent of Azerbaijan, including the Nagorno-Karabakh region and seven surrounding districts.
The two countries signed a ceasefire agreement in 1994. The co-chairs of the OSCE Minsk Group, Russia, France and the US, are currently holding peace negotiations. Armenia has not yet implemented the UN Security Council's four resolutions on withdrawal of its armed forces from the Nagorno-Karabakh and the surrounding districts.
Latvian Socialist MEP Andrejs Mamikins says the upsurge in violence between the two sides shows that the ENP has "hit a dead end."
The eastern partnership programme envisages political association and economic integration of the EU with Azerbaijan, Armenia, Belarus, Georgia, Moldova and Ukraine. It does not envisage EU membership.
The ENP countries had pledged to support only the EU's calls for peace, Mamikins said.
"As we hurt Azerbaijan, it has broken off all contacts in the framework of inter-parliamentary cooperation. The policy hit a dead end after Ukraine, Georgia and Moldova signed an association agreement with the EU," Mamikins argues.
The MEP, who is a Vice-Chair of Parliament's delegation for relations with Belarus, said a new ENP should be developed, stressing the importance of normalising relations with Azerbaijan.
"First of all, let's apologise to Azerbaijan. Let's say that, yes, there are problems with human rights and freedom of the media in Azerbaijan, but we, the European Parliament, said this to Baku in a disrespectful manner. We have to think of something new for Baku.
"What does Azerbaijan mean to the European Parliament? Does it merely mean an inexpensive source of gas? Or do we consider Azerbaijan an important actor and trust in this country with regard to issues like countering terrorism?
"We, EU countries, are unable to define what we want from ENP countries. Nor are able to give post-Soviet countries something in the current form of eastern partnership," he added.
His comments are partly echoed by Latvian ALDE group MEP Iveta Grigule, who recently returned from Baku where she held talks on the Nagorno-Karabakh conflict.
Grigule, a member of the foreign affairs committee, also raised the issue of Azerbaijan's return to the Euronest parliamentary assembly.
"The Euronest parliamentary assembly is discussing cooperation with Azerbaijan, and it means that Azerbaijan should participate in these discussions, and not remain on the sidelines," she said, adding that it is necessary to talk about the problem the country has, and sooner or later, Azerbaijan's voice will be heard.
The MEP also touched upon the anti-Azerbaijani resolution adopted by the European Parliament last year, calling it a mistake.
The resolution adopted last September called on the European Commission "to review and suspend temporarily, if needed, all funding not related to human rights, civil society and grassroots level people-to-people cooperation granted to Azerbaijan through the European neighbourhood instrument" and recalls Parliament's decision to send a delegation to Azerbaijan.
"Azerbaijan and the EU should continue to work, despite such incidents," said Grigule. "Latvia is a small country, too, and we are sometimes also criticised by the EU. We understand how it hurts, because we are also doing our best, as well as Azerbaijan is."
Their comments come in the wake of a recent visit to Baku by EU foreign affairs chief Federica Mogherini.
The Strasbourg-based Council of Europe entered the fray this week when it voiced "real concern" at a serious increase in new hostilities in the Nagorno-Karabakh conflict.
Pedro Agramunt, President of its parliamentary assembly, said he was "extremely worried" at the upsurge in violence which has cost an estimated 35 lives, including 12 Azerbaijani servicemen, in recent days.
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".
Further comment on the issue came from Azerbaijan MP Azay Guliyev, who said the "continuous unlawful deployment of Armenian armed forces on the occupied territories of Azerbaijan remains the main reason of the deterioration of situation and cause serious threat to the peace and stability in the region."
He said the UN resolutions "condemn the use of force against Azerbaijan and the occupation of its territories as well as once again affirm the sovereignty, territorial integrity and inviolability of internationally recognized borders of Azerbaijan."
The MP said, "Azerbaijan calls for the OSCE to demand from Armenia to eliminate the tension created in the front line and stop the illegal occupation of Azerbaijani territories in accordance with the requirements of UN Security Council resolutions."