Akmal Shaikh was arrested in north-west China in 2007
British consular staff are set to fly to the Chinese region of Xinjiang to see condemned Briton Akmal Shaikh and discuss his case.
The government has asked for clemency to be shown because he is believed to suffer from bipolar disorder.
The Beijing-based staff will fly to the city of Urumqi where he is being held.
Mr Shaikh, 53, from London, is due to be executed on Tuesday. He has denied knowledge of the 4kg of heroin found with him in 2007.
The Briton would become the first EU national to be executed in China in 50 years.
CHINA DEATH PENALTY
China executed 1,718 people in 2008, according to Amnesty International
Last year 72% of the world's total executions took place in China, the charity estimates
It applies to 60 offences, including non-violent crimes such as tax fraud and embezzlement
Those sentenced to death are usually shot, but some provinces are introducing lethal injections
His family says he has bipolar disorder and travelled to China with the promise of being made into a pop star, despite having no singing experience.
The charity Reprieve, which promotes human rights, said he was then duped by a criminal gang into unwittingly carrying drugs for them.
They say his strange behaviour was "influenced or caused by" his mental illness.
Consular staff are expected to talk to Chinese officials about the case.
Reprieve said two members of Mr Shaikh's family - his cousins Soohail Shaikh and Nasir Shaikh - would fly to Urumqi on Saturday.
There they will deliver legal petitions asking for his case to be reviewed.
Soohail Shaikh says in his petition: "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."
Reprieve said Mr Shaikh's execution had been set for 29 December.
Mr Shaikh's final appeal against his conviction was turned down earlier this week.
The UK has raised the case with China 10 times during the last six months at senior levels to argue forcibly against the use of the death penalty.
But so far China has resisted calls to stop the execution.
The BBC's Michael Bristow in Beijing said: "British officials appear to be making a last-ditch effort to stop Akmal Shaikh's execution."
The Briton's brother has already made a plea for mercy.
In a letter to the Chinese Ambassador, Akbar Shaikh said his brother's life had been destroyed by mental illness.
Mr Shaikh said his brother, when healthy, was a kind and harmless man who was much loved by his family.
He said: "I beg you to spare his children the trauma of losing their father, and to spare me the agony of losing my brother.
"Akmal's cousins are also horrified by the prospect of his death, so far away and without the possibility of being able to say goodbye.
"Perhaps the most important of all is the effect this would have on my mother. She is a frail woman, and our family have not been able to break the news to her that she may lose her youngest child next week."
UN special rapporteur Philip Alston said there were "strong indications" Mr Shaikh suffered from mental illness, which the Chinese courts had failed to take into account.
"Executing a mentally ill man would be a major step backwards for China and I very much hope that the government will grant clemency in this case," he said.
And Downing Street said the prime minister had written to China's leaders to express his dismay after Mr Shaikh's sentence was upheld by the Supreme People's Court.