Jasmine Rana has travelled from Edinburgh to present the 70,000-signature petition demanding action in the case of 70-year-old Mohammad Asghar who was convicted of blasphemy.
The grandfather, who has mental health problems, was the target of a shooting in Adiala prison in Rawalpindi last month.
His family, backed by the charity Reprieve, say his safety cannot be assured in Pakistan and he should be returned to the UK.
First Minister Alex Salmond said he is “extremely concerned” about Mr Asghar’s welfare and said he would support a prisoner transfer agreement following a meeting with Ms Rana on Wednesday.
Family solicitor Aamer Anwar said: “It is no longer acceptable for the UK Government to hide behind a wall of silence whilst Mr Asghar’s life hangs by a thread and we call upon the Prime Minister to personally and publicly hold the Pakistani government to account.
“Every minute that passes increases the risk to Mohammad Asghar’s life.”
Mr Asghar was sentenced to death in January after writing letters to a number of people claiming to be the Prophet Mohammed.
He was arrested in 2010 following a blasphemy complaint by a tenant with whom he was having a dispute.
The Scot had previously been diagnosed as suffering from severe paranoid schizophrenia but this was not taken into account by the authorities during his trial, Mr Anwar said.
A Downing Street spokeswoman said this week that the Government has repeatedly raised Mr Asghar’s case “at senior levels” and is working to ensure that he is receiving the best possible support.
She said the Prime Minister had asked the minister for Pakistan Tobias Ellwood to meet Mr Asghar’s family today.