With continued backing from Glasgow solicitors Harper Macleod, the instructors are to ask the Lords to overturn the decision of Scotland's top judges to reject their claim for parity with teachers under the Equal Pay Act.
The instructors, seven of whom are women, are likely to claim indirect discrimination. They say they were not paid the same as teachers, although they were doing similar work, a point accepted by the employers.
An industrial tribunal in Glasgow, followed by the Employment Appeal Tribunal in Edinburgh, backed their case. The 12 councils in the former Strathclyde Region then took an appeal to the Court of Session.
Scotland's most senior judges supported the councils' arguments that teachers' pay was bound by a statutory framework and that the pay of other local authority staff was not. Teachers had higher qualifications and longer training.
Rod McKenzie, the instructors' solicitor, hoped the case would be heard - although there is no automatic right to appeal - and is optimistic the verdict would support the two earlier tribunals. The bill could run to more than pound;100,000.
Councils stand to lose nearly pound;5 million in back pay. More than 130 cases are lodged at the tribunal in Glasgow.
Linda Marsh, the councils' spokeswoman, said in October that the Court of Session verdict showed that the case had nothing to do with sex discrimination and was "simply a ruse whereby people are attempting to circumvent the regrading system". Staff were trying to win large pay rises that councils could not afford.