The Cuban government has accused church leaders of being mercenaries and counter-revolutionaries by not giving their overwhelming support for the new document.
If introduced, the constitution would re-introduce the role of a prime minister, limit the president to two five-year terms, protect private property and establish discrimination laws. State media have largely supported 'yes' to adopting it.
Clergy have been less enthusiastic because of a fear that it will weaken the rights to freedom of religion.
Many Church leaders have reported receiving phone calls threatening them with arrest and imprisonment if they do not instruct their congregations to vote 'yes' in the vote on 24th February.
On Wednesday, the president of the Western Baptist Convention received a call from the Deputy Head of the Office of Religious Affairs, Sonia García García, who belongs to a wing of the Cuban Communist Party.
She told him:"from now on Carlos Sebastián [the president's colleague, the General Secretary] will no longer be treated as a pastor, but as a counter-revolutionary."
After that, General Secretary Rev. Carlos Sebastián Hernández Armas told the religious freedom charity Christian Solidarity Worldwide: "I have total confidence that, even in Cuba, God reigns. Pray for me and my family...my wife and I have spoken and prayed because at any moment they could take me prisoner."
Meanwhile, a Baptist pastor in Baracoa, Revd Michel García Avilés, has had his house surrounded by state security and police all week.
He said government officials called him a "mercenary" in a meeting on Sunday and threatened him with arrest.
A pastor who has been outspoken about his intention to vote 'no', and whose church belongs to the Cuban Council of Churches, which has historically had a mutually supportive relationship with the government, was told to "take a vacation," and was suspended for 15 days after he refused to do so.
On Thursday, the leader of another major Protestant denomination received a phone call from García García who accused the denomination of telling its members to vote 'no'.
He said: "I told her clearly that they could not count on our vote for a constitution that does not represent us. I told her that they did not listen to our demands, they did not give us an opportunity to discuss, they refused us permissions, they accused us of being fundamentalists and backwards. The church has had no rest in these days. They are besieging and intimidating us just for defending our rights and our principles. They cannot count on us now. After these words she hung up on me. Please pray for us."
It's reported that students and workers across the country have been summoned and told to vote 'yes' and that Christians have been 'screamed insults at', according to Christian Solidarity Worldwide.
At least one person was dismissed from their job as a result.
CSW's Americas Team Leader Anna-Lee Stangl said: "The Cuban government must immediately cease its attempts to intimidate and pressure religious leaders and their congregations as the country prepares to vote on Sunday.
"As church leaders have repeatedly pointed out, the government largely ignored their requests and recommendations during the period of national consultation on the draft constitution specifically with regards to freedom of religion or belief and freedom of conscience. It is telling that the Cuban government considers calling for freedom of religion or belief a 'counter-revolutionary' activity. We call on the Cuban government to cease all harassment and pressure tactics and allow people to vote according to their conscience on 24 February."
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