It wants the pontiff to condemn the occupation of Nagorno-Karabakh and surrounding Azerbaijani provinces and make the country take public responsibility for the Khojali massacre in which more than 600 people were killed in 1992.
President of the Association for Civil Society Development in Azerbaijan, Elkhan Suleymanov, wrote to the leader of Catholics all over the world.
He said: "We wish to express our profound concern regarding potential issues that might be addressed during your upcoming visit to Armenia, including misleading efforts by Armenian authorities to justify their illegal occupation of Nagorno-Karabakh and other adjacent Azerbaijani territories."
Around 20% of Azerbaijani territory was occupied by Armenia during the post-Soviet power vacuum in the early 1990s.
In the letter, Mr Suleymanov stressed that UN resolutions are calling for the immediate and unconditional withdrawal of Armenian troops from the territories.
The White House issued a statement recently that emphasised the importance of "the return of the territories surrounding Nagorno-Karabakh to Azerbaijan's control."
Elkhan Suleymanov continued in his letter to Pope Francis: "In light of this, we wish to appeal to the spiritual and moral responsibility bestowed upon the Holy See and to call upon Your Holiness to address in Yerevan the tragedies that have taken place in Nagorno-Karabakh, and to urge Armenian authorities to comply with the conclusions drawn by the international community, in particular the return of all occupied Azerbaijani territories in accordance with United Nations and Council of Europe resolutions."
He also referred to the close relations between Vatican and Azerbaijan, saying the country is very tolerant: "We are certain Your Holiness is well aware of the deep-rooted tradition of religious and ethnic tolerance that has guided the people of Azerbaijan for many centuries, leading to the peaceful co-existence of Christians, Jews and Muslims."
Pope Francis will arrive in Armenia for a three day visit on Friday.
This year, he has already travelled to Mexico, the USA and Cuba.