In early 2001, representatives of Armenia, Azerbaijan, France, Russia and the United States met in Paris and in Key West, Florida.

On 7 October 2002 during the CIS summit in Chișinău, the usefulness of the Minsk Group in peace negotiations was brought up for discussion by both the Armenian and the Azerbaijani delegations.

Ilham Aliyev in his interview to the local media on January 12, 2022 said that after 30 years of experience, the Minsk Group co-chairs "are on the verge of retirement" and therefore "he wishes them good health and a long life".

Azerbaijan's foreign minister Jeyhun Bayramov also noted that interaction between the co-chairs of the OSCE Minsk Group has been completely paralyzed and that the peace process cannot be held hostage to and be guided by a non-existent format.

Ambassador to Azerbaijan, the EU would make more sense as a co-chair, because it would represent more of Europe and has experience mediating similar conflicts in the Balkans.

He suggested that the Minsk Group redefine its mission, e.g. by enabling reconstruction to its approved mandate, otherwise it may continue as "an intriguing backwater of international diplomacy".

[26] According to Carey Cavanaugh, another former US co-chair of the Minsk Group, the organization’s consensus-based decision-making process and its rotating leadership rendered it “structurally flawed” to act as a peacemaker, and the United Nations would have been a better option to facilitate peace.

[27] For analyst Laurence Broers, the Minsk Group’s future remains unclear, with its failure caused by factors like the normative ambiguity of its attempts to balance the countervailing principles of self-determination and territorial integrity; its secretive, narrow and top-down modus operandi; and its default to performative over substantive diplomacy since 2011, when occasional summits in far-away capitals with little or no interaction in between made the peace process alien to Armenian and Azerbaijani societies.

[30] During the 2020 Nagorno-Karabakh War, France received particularly harsh criticism in Azerbaijan,[31] to the point of being viewed as "unworthy" to hold the position of the OSCE Minsk Group co-chair.