Six people died when Mr Clarke lost consciousness at the wheel of the vehicle in Queen Street in December 2014.
A Fatal Accident Inquiry (FAI) held last year heard that the driver had a history of health issues – including a previous blackout in 2010 when at the wheel of a stationary bus – but had not disclosed his medical background to his employers or the DVLA.
The Crown Office has consistently said that there is insufficient evidence in law to raise criminal proceedings against Mr Clarke but the families of the victims disagree and announced their intention to stage a private prosecution during the FAI.
Lawyers for the family of Jack and Lorraine Sweeney and their granddaughter Erin McQuade delivered a Bill for Criminal Letters to the Crown Office on Wednesday and hope it will lead to a private prosecution.
An application for public funding has also been made to the Scottish Government.
The rare bill seeks the agreement of Lord Advocate Frank Mulholland QC to pursue the case but lawyers said it can still continue and be ruled on by High Court judges without his approval.
A statement from the Sweeney and McQuade family lawyer said: “Paul Kavanagh, Gildeas solicitors, intimates on behalf of the relatives of Jack and Lorraine Sweeney and Erin McQuade that a Bill for Criminal Letters was delivered to the Lord Advocate.
“We have sought the concurrence of the Lord Advocate and look forward to receiving a response within seven days.
“This is the initial process that the family hope ultimately will lead to the prosecution of Henry (Harry) Clarke in the criminal courts.”
A spokesman for the Crown Office confirmed the Bill for Criminal Letters had been received and will be given “due consideration”.
He added: “The Crown position on this will be made clear to the families and the court when appropriate.”
A Scottish Government spokeswoman said: “Scottish ministers have received further information in support of an application for public funding towards a private prosecution from lawyers acting on behalf of the families of the Glasgow Bin Lorry tragedy, and will now consider this closely.”
Last month, Mr Mulholland insisted it would have been “wrong” to prosecute Mr Clarke.
He said he knew the decision not to charge the 58-year-old was “not a popular one”, adding he was aware of the feelings of the victims’ families on the matter.
The sheriff who oversaw the FAI ruled that the accident might have been prevented if the Mr Clarke had “told the truth” about his history of blackouts.
Sheriff John Beckett QC found Mr Clarke “repeatedly lied in order to gain and retain jobs and licences”, and “deliberately concealed relevant information from the DVLA”.
Stephenie Tait, Jacqueline Morton and Gillian Ewing also died in the tragedy which was marked with a memorial service in Glasgow last month.
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