Akmal Shaikh, 53, from Kentish Town in north London, is due to be put to death at 10.30am on Tuesday (2.30am GMT) after being convicted of smuggling heroin.
The Chinese have said knowledge of his execution is being withheld from the prisoner on "humanitarian grounds", according to human rights charity Reprieve.
His cousins Soohail and Nasir Shaikh, from London, were due to arrive in Urumqi in north-west China, where he is being held, this afternoon to try to prevent the execution.
They were being joined by British embassy officials to deliver pleas for clemency to Chinese president Hu Jintao and the Chinese courts.
They were also hoping to become the first family members to meet with their relative in two years.
Father-of-three Mr Shaikh was arrested in Urumqi in September 2007 and charged with drug smuggling.
After being convicted he lost a final appeal last week, but campaigners claim his mental illness - bipolar disorder - has not been taken into account.
Clive Stafford Smith, director of Reprieve, which helped arrange visas for the family members, said: "The timing of Akmal's notice of his own execution raises obvious concerns.
"We hope that the Chinese authorities have kept him in the dark that his execution is only hours away because they are going to show clemency.
"Only then would it truly be humanitarian for him to be the only person in the world not allowed to know.
"I have been in constant contact with Akmal's family, and they are simply praying for a reprieve, fearing for the health of his mother, who is very frail."
The relatives were hoping to visit their cousin tomorrow morning, when he is also due to hear his fate for the first time.
In his petition, Soohail says: "We plead for his life, asking that a full mental health evaluation be conducted to assess the impact of his mental illness, and that recognition be made that he is not as culpable as those who might, under Chinese law, be eligible for the death penalty."
Akbar Shaikh, Akmal's brother, also says in a letter from the family to the Chinese president: "We plead for mercy and clemency.
"We are not asking for special treatment for Akmal because he is British, but simply as a family who are devastated at the possibility of losing our son, our brother, our father, our cousin."
Reprieve has said it has medical evidence that Mr Shaikh was suffering from a delusion that he was going to China to record a hit single.
Once there he was duped by a criminal gang into unwittingly carrying drugs for them.
Prime Minister Gordon Brown is among those who have urged the Chinese government to grant Mr Shaikh a reprieve from the death sentence.
British consular staff have also flown to the Chinese region of Xinjiang to see the condemned Briton and discuss his case with local officials.
If the sentence is carried out, it would be the first time an EU national has been executed in China for 50 years.
A Foreign Office spokeswoman said: "We have made representations at the highest possible levels.
"The Prime Minister wrote to the Chinese government on December 21 and the Foreign Secretary has also written.
"We have made our position to the Chinese authorities quite clear.
"The Prime Minister, ministers and other officials have been and remain closely engaged."
Stephen Fry is among those who have appealed for mercy.
The television presenter, who also suffers from bipolar disorder, yesterday Tweeted: "Surely time for one last push urging the Chinese government to show clemency for Akmal Shaikh."
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