Mr Asghar, who has paranoid schizophrenia, was shot by a prison guard in September. He has since been recovering in hospital.
His relatives urged Mr Salmond to do all he could to secure his release.
A petition calling on the prime minister to personally intervene has gained more than 68,000 signatures.
Mr Asghar's family will present the petition to Downing Street on Friday.
Following the meeting, Mr Asghar's solicitor Aamer Anwar said: "We welcome the unconditional support of the Scottish Government, the First Minister and their repeated attempts to assist Mr Asghar and his family.
"We are disappointed that David Cameron has yet to respond to the request to meet with him on Friday when we hand in 70,000 signatures on a petition to Downing Street.  All supporters of the petition will be asked tomorrow to email David Cameron seeking his urgent personal intervention.
"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 call upon the Prime Minister to personally and publically hold the Pakistani government to account."
He said direct requests to meet with Mr Cameron had so far been refused. Instead, Mr Anwar will meet with Foreign Office Minister Tobias Ellwood on Friday.
Pakistani authorities have attempted to return Mr Asghar, who is from Edinburgh, from hospital to the same prison where he was attacked.
On 3 October, local doctors said he was too unwell to return to jail.
Charity Reprieve said that that the authorities in Punjab province, where Mr Asghar is being held, have refused requests by Mr Asghar's lawyers to access the results of an investigation into the shooting, as well as information relating to his medical assessments since the incident.
The information that has been denied "could support arguments that Mr Asghar's sentence is illegal on the grounds of his mental illness", according to Reprieve.
Kate Higham, an investigator at Reprieve said: "Mr Asghar is an ageing, seriously ill man who should never have been sentenced in the first place. The British government must redouble its efforts to ensure that Mr Asghar is returned home to his family in Edinburgh, before it's too late."
Mr Asghar was sectioned under the Mental Health Act in Edinburgh's Royal Victoria Hospital shortly before his trip to Pakistan in 2010.
He was arrested there for writing several letters claiming to be a prophet and was sentenced to death for blasphemy.
Those accused of blasphemy in Pakistan are at high risk of attack from religious extremists.
First Minister Alex Salmond has met the family of Mohammad Asghar, the Scot imprisoned on death row in Pakistan over blasphemy allegations.