Mohammed al-Nimr says the family is "extremely worried" after Ali al-Nimr, 20, was moved to a prison in the Saudi capital, Riyadh, a month ago.
Last week, Ali was able to call his family for the first time since being transferred from Dammam, 400 kilometers (250 miles) away.
He told them he was being held in solitary confinement, in a block housing about 30 other death row prisoners.
"We're forbidden from visiting them," Mohammed al-Nimr told CNN. "Anything is possible now in regard to their fate."
He said the Saudi authorities may not even tell family members before carrying out his son's execution, leaving them to find out publicly, after his death.
"As soon as King Salman signs the execution order, it can be implemented at any moment," he told CNN. "Usually the Interior Ministry does not notify anyone that they will kill their child ... at a specific time.
"We could at any time turn on the TV or the radio and hear the decision announced there."
Ali al-Nimr was 17 and still in high school in 2012 when he was arrested for taking part in protests calling for social and political reforms in Saudi Arabia's restive Qatif province.
A court later convicted him of charges including belonging to a terror cell, attacking police with Molotov cocktails, incitement and stoking sectarianism, according to state media.
Reprieve, an NGO which campaigns against the death penalty, says Ali al-Nimr was "arrested without a warrant
... held in pre-trial detention for two years ... tortured and forced to sign a false confession. This was the only evidence brought against him."
But his final appeals have been rejected and he now faces beheading, along with the additional, rarer punishment of "crucifixion," which would see his body placed on public display as a warning to others.
CNN has asked the Saudi government for comment on the case but has received no response.
Mohammed al-Nimr insists his son has done nothing to deserve such a "harsh and unjust verdict," which he says "does not fit the boy, does not fit international law, (does not fit) the laws of the kingdom of Saudi Arabia.
"Even if all the charges against Ali were proven, he still would not deserve to be killed, within Saudi law. In my opinion the charges were not proven because the trial was flawed," he said.
"Death is still death, but this ugly scenario, and this ugly death, for someone who was a child when he was arrested, is something we completely and utterly reject."
Al-Nimr says a friend of his son, Dawoud al-Marhoon, who, like Ali, was 17 when he was arrested on similar charges, has also had his conviction upheld; he too faces beheading -- though not crucifixion.
Ali's father believes the charges against his son were motivated by the fact that his brother, Ali's uncle, is the firebrand Shiite cleric Sheikh Nimr al-Nimr, who has also been condemned to death by a Saudi court.
Nimr al-Nimr's sentence of beheading and crucifixion has been upheld, and like his nephew he is now on death row.
His brother says the punishment could be carried out at any time, without the family being warned in advance, once the King signs his death warrant.
Mohammed al-Nimr was himself jailed for 26 days last year, for posting tweets from his brother's trial.
"Ali went out in demonstrations just like tens of thousands of people went out to protest in the region during the Arab Spring," his father said. "... But the charges that they stuck (Ali) with, I think, were spiteful charges. It was spite against his uncle, Sheikh Nimr al-Nimr."
CNN has also asked the Saudi government for comment on the uncle's case, but has received no response.
Whatever happens to his son, Mohammed al-Nimr has appealed to the family's supporters not to resort to violence.
"Even if -- God forbid -- something happens to Ali, we don't want anyone to ... commit any act that is not peaceful. Our demands remain peaceful and our objections have also been peaceful."
U.N. experts and human rights organizations have urged Saudi authorities to call off the sentence
, on the grounds that Ali al-Nimr was a minor at the time of his alleged crimes.
Activists say if Ali is executed it would violate the U.N. Convention on the Rights of the Child, to which Saudi Arabia is a party.
The European parliament has passed a resolution calling on the Saudi King "to halt the execution
of Ali Mohammed al-Nimr and grant a pardon or commute his sentence."
And world leaders including French President Francois Hollande have appealed to Saudi Arabia
-- a key Middle East ally -- to show clemency.
British opposition leader Jeremy Corbyn has called on Prime Minister David Cameron to take action
over the case; after "speaking privately" to authorities in Saudi Arabia, Britain's foreign secretary said he did not think the execution would go ahead
"I do not expect Mr. al-Nimr to be executed," Philip Hammond told the UK parliament last week.
But Mohammed al-Nimr says his son's fate rests in the King's hands.
"We are of course worried. I am worried and the families of the other prisoners in the kingdom are also worried. ... I believe the real solution lies in the hands of King Salman."