Ali Mohammed Baqir al-Nimr, convicted of crimes including breaking allegiance to the country's king, could be beheaded and his body displayed in public, Labour's leader said.
He called on the PM to scrap a UK bid to provide services to Saudi prisons.
The Foreign Office said it would raise the case "urgently".
Mr al-Nimr was accused of taking part in anti-government protests in the eastern part of the country in 2011, involving the country's Shia Muslim minority, and was arrested the following year.
State media later reported he was found guilty of a long list of crimes including sedition, breaking allegiance to the king, rioting, using petrol bombs against security patrols, robbing a pharmacy and more. He was 17 at the time of his arrest.
In a letter to Mr Cameron, Mr Crobyn wrote: "As you may be aware, Ali has now exhausted all his appeals and could be executed any day - in a particularly horrific manner, which involves beheading and the public display or "crucifixion" of the body."
The Labour leader said Mr al-Nimr's trial contained "numerous fair trial violations".
He urged Mr Cameron to raise the case "directly" with his Saudi counterparts and "request that they commute the unjust and horrific sentence... which violates any number of international laws".
Mr Corbyn also raised questions about a Ministry of Justice bid to provide services to Saudi prisons.
He referenced reports Justice Secretary Michael Gove wanted to withdraw the bid but was "blocked" by "other departments".
And he wrote: "Will you step in to terminate the Ministry of Justice's bid to provide services to the Saudi prisons system - the very body, I should stress, which will be responsible for carrying out Ali's execution?"
The Labour leader concluded: "Ali's case is especially urgent - the secrecy of the Saudi system means that he could face execution at any time, and even his family may only find out after the event.  There is therefore no time to spare in taking this up with the Saudi authorities, if we are to prevent a grave injustice."
A Foreign Office spokesman said: "We understand that Ali Mohammed Baqir al-Nimr's legal process has finished and his final appeal has been denied. We will raise this case urgently with the Saudi authorities.
"The abolition of the death penalty is a human rights priority for the UK. The UK opposes the death penalty in all circumstances."
David Cameron should urge Saudi Arabia to commute the sentence of a protestor who could be executed there "any day", Jeremy Corbyn says.