They took their case for the 59-year-old to face a trial to the Appeal Court in Edinburgh earlier this year.
The rare move followed a controversial decision by the Crown Office not to prosecute Mr Clarke over the 2014 tragedy.
Judges Lady Dorrian, Lord Menzies and Lord Drummond Young have been considering the Bill for Criminal Letters since hearing final arguments in the case in October and will issue their decision on Friday.
The case was brought by relatives of crash victims Jack and Lorraine Sweeney, aged 68 and 69, and their granddaughter Erin McQuade, 18.
Stephenie Tait, 29, Jacqueline Morton, 51, and Gillian Ewing, 52, also died in the collision, which happened after Mr Clarke blacked out behind the wheel in the city centre on December 22 that year.
The court is also expected to rule on a similar plea for a private prosecution of motorist William Payne, lodged by the families of students Mhairi Convy and Laura Stewart, who were knocked down and killed in Glasgow in 2010.