Akmal Shaikh, aged 53, who campaigners say has a mental illness, was due to be put to death at 2.30am GMT for smuggling 4kg of heroin, but was unaware of his death sentence until yesterday.
Campaigners believe Shaikh was duped into transporting the drug.
Sally Rowen, legal director of the campaign group Reprieve, said: “He has been told this morning that he is to be executed.
“I do not know what his reaction was but I just know that he has been told.
“The Chinese authorities had always said they would tell him 24 hours ahead.”
She said the authorities had in the past given reprieves “right at the last minute”. “There is no reason to think that this might not happen in this case,” she said.
Two of his cousins travelled to China to make representations to the authorities, but his daughter Leilla Horsnell said she was not optimistic.
She told the BBC: “I’d like to be hopeful, but time just seems to be running out.”
The Chinese only informed Shaikh of his fate 24 hours before it was due to happen on humanitarian grounds. Horsnell said she thought this was a good thing “because I don’t even think he would understand because we don’t know how much his mental state has deteriorated”.
She said: “I don’t think him being told would mean anything ... if anything, it might make it worse if he was aware of what was happening.”
His cousins Soohail and Nasir Shaikh, from London, flew from Beijing to Urumqi in north west China on Sunday.
They joined two British embassy officials to deliver pleas for clemency to Chinese president Hu Jintao and the Chinese courts.
Rowen said they were allowed access to Shaikh and were the first family members to have face-to-face contact in two years.
Father-of-three Shaikh, from Kentish Town in north London, was arrested in Urumqi in September 2007 and charged with drug smuggling. He lost a final appeal last week, but campaigners claim his bipolar disorder has not been taken into account.
A vigil was taking place in central London yesterday outside the Chinese embassy.
A spokesman for the Chinese embassy in London said Shaikh was found with more than 4kg of heroin, which he said was enough to kill 26,800 people.
According to Chinese law, he said, being caught with 50g of heroin is enough for the death penalty to be applied.