The referendum is due to happen on 24th February.
High ranking communist party officials summoned Christian, Yoruba and Masonic leaders in Santiago on Tuesday for assurance that they and their congregations would all be voting to adopt the new constitution.
Religious leaders elsewhere in the country have told the freedom charity Christian Solidarity Worldside (CSW) that they had been summoned to similar meetings in the past week.
They are concerned about supporting it as the draft looks like it could significantly reduce protections for freedom of religion or belief (FoRB) and it weakens references to freedom of conscience.
One church leader told CSW that "all [the Communist Party of Cuba leaders] wanted was to intimidate us because they thought that we were going to openly promote a no vote. In my eyes, it was a meeting to attempt to pressure us, I think they are very worried."
These meetings occured after several church denominations have criticised the new constitution openly.
The Eastern Baptist Convention, the Methodist Church of Cuba and the Assemblies of God, as well as from the Cuban Catholic Bishops Conference have all said they have concerns about freedom of conscience potentially being taken out.
The Catholic bishops wrote: "we regret the absence of recognition of the diversity of political opinion (see article 40), as well as the need to make explicit that the right to life (article 43) must be respected from the moment of the individual's conception until the natural death and, in turn, it would be desirable to include the exclusion of the death penalty.
"In the same way, a greater clarification on the right to conscientious objection is required (art. 59), so that the preservation of one's right does not imply violating the right and conscience of others. The causes, scope, means and limits in relation to art. 3 related to the defense of the Homeland, to protect the integrity and the exercise of the right of the people
In a cross-denominational petition, they ask for the following ammendments:
"The Cuban State guarantees the churches and religious institutions the freedom to manifest their religion or belief and to do so individually and collectively, in public and in private, and the right to teach and practice the corresponding worship and to independently observe their principles and faith."
"The Cuban government, its bodies and authorities will abstain of interfering with the internal life of religious associations, and of creating organizations to control them."
The Assemblies of God posted their position on Facebook, saying: "we found that none of the articles proposed for modification by the Cuban church (16 articles) were considered; modifications that, in addition, many of them coincided with the criteria expressed by the Cuban people. For this reason, this Constitution does not represent us and exclude us."
They also express concern that the new constitution proposes a subtly gender ideology which they say is: "violates the principles of the Word of God by distorting the original design of the family".
CSW say their sources were asked directly how they and their members planned to vote and that two members of the Communist partytold religious leaders on Tuesday: "that the enemy was trying to destroy the Revolution, that a unanimous 'yes' vote was needed and that they would do everything in their power to ensure that things went well."
Mariela Castro Espín, the daughter of Raul Castro and a member of the National Assembly, referred to the Catholic Church recently as "the serpent of history" and called for a strong state response to the Catholic Bishops Conference statement.
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