The Sweeney/McQuade family want Mr Clarke prosecuted over the December 2014 crash which killed six people.
The families of students Mhairi Convy and Laura Stewart, who were knocked down and killed in Glasgow in 2010, are seeking to prosecute William Payne.
Judges at the High Court in Edinburgh have begun hearing submissions.
Both Mr Clarke and Mr Payne blacked out at the wheel of the vehicles they were driving.
The Crown Office decided not to prosecute either on the basis that they did not commit crimes because they did not know that their medical conditions would have caused them to pass out.
However, the relatives of those who died in the collisions believe that the Crown made the wrong decision.
They now want to bring their own prosecution against the two men and have gone to the High Court to seek permission to start proceedings against the two men.
Much of what was said during Tuesday's hearing, held before judges Lord Carloway, Lord Bracadale and Lady Paton, cannot be reported for legal reasons.
Bereaved families have launched legal bids to bring private prosecutions against Glasgow bin lorry driver Harry Clarke and another fatal crash driver.