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.

Seven-year-old Lily-Grace Hooper was told she could not use her cane at Hambrook Primary School in Winterbourne Down near Bristol.
Her mother Kristy Hooper said she was "devastated" when she was told of the decision last Thursday.
The school stressed the measure was a temporary one while the situation was discussed with the family.
Ms Hooper, who first spoke to the Bristol Post, said her daughter suffered from severe sight impairment after she suffered a stroke at four days old.
She had been using the lightweight cane in school since April without issue, her mother added.
"When I picked her up on Thursday her teacher said... Lily-Grace could no longer bring her cane into school. I was quite shocked," Ms Hooper said.
"Lily-Grace is quite sad because she likes to use her cane because of the independence it gives her."
Head teacher Jo Dent said: "The pupil has not been banned from bringing in their cane, we have simply asked them to not use it around school as a temporary measure until we have the chance to meet with the parent and discuss the situation.
"It was initially hoped we would have this resolved within a day or two.
"The school's mobility officer raised health and safety issues around the new cane following a recent risk assessment.
"We have to consider all of our pupils, so it is important that we have an opportunity to discuss the situation before we make any decisions.
"We are very keen to resolve this issue as soon as possible and have been actively seeking to engage with the parent to bring this to an agreeable conclusion."
Geoff Cox, from the Health and Safety Executive, said there was nothing in HSE regulations that would stop a child "from using a walking stick in school".
He said it was up to those involved to work out "sensible" arrangements.
"I hope common sense prevails here," he said.
First Minister Alex Salmond has met the family of Mohammad Asghar, the Scot imprisoned on death row in Pakistan over blasphemy allegations.
[NEXT_CONCEPT]
A blind schoolgirl has been asked to stop using her white cane in school due to health and safety concerns.