Mohammad Asghar, of Edinburgh, was arrested in 2010 in Rawalpindi, near Pakistan's capital, Islamabad, for claiming to be the Prophet Mohammad.
Relatives say he suffers from mental illness and was treated for paranoid schizophrenia in Edinburgh before returning to Pakistan in 2010.
He was convicted last week but his family are appealing for him to be released from custody to receive medical help.
The blasphemy complaint was brought against Mr Asghar by a tenant with whom he was having a dispute.
Lawyers said they will appeal against the conviction and are worried about his mental condition and physical safety while he is in prison.
In a statement, Mr Asghar's family said: "Our father, a British citizen, has been sentenced to death in Pakistan for blasphemy despite being a confirmed paranoid schizophrenic by the doctor who was treating him in the UK.
"We, his family, want him released by the Pakistani government so he can be treated appropriately for his medical condition.
"As a result of a property dispute with one of his tenants, my father was jailed pending a trial."
The family complained that the hearing dates kept being changed meaning that by the time of Mr Asghar's trial, he had already been subjected to "horrific" jail conditions in a cell shared with several other men for three years.
The family added: "Throughout this time he had minimal access to medication that might have helped his mental illness for three years.
"We are really upset and concerned that they will never release him and that he will die in jail.
"We just want him back home where hopefully he can be treated for and recover from his mental illness.
"We urge the British Government to do everything they can to make sure that he is safe."
A petition has been launched on change.org addressed to Prime Minister David Cameron and Scotland First Minister Alex Salmond calling for Mr Asghar's release.
Scores of people have been arrested in Pakistan under the country's blasphemy laws but death penalties are rarely carried out, it is claimed. Few leaders have shown willingness to tackle the issue since two prominent politicians who criticised the blasphemy law were murdered in recent years.
Maya Foa, director of human rights charity Reprieve's death penalty team, said: "It is an absolute disgrace that Mr Asghar's lawyers are not being allowed into the prison to see their mentally-ill client just after he has been given a death sentence.
"We are calling on the British and Pakistan governments to take all possible measures to ensure that Mr Asghar is protected and that his lawyers are given immediate access to the prison."
The Scottish Government has said it strongly opposes the death and, as a "minimum", urged the Pakistani authorities to abide by the moratorium they have on the death penalty.