The Enquirer first reported in June 2015 that newly installed barriers between junctions 28 and 31 were being cut out and replaced.
Contaminated material supplied by Tarmac was blamed for the problems and the case is now going through the courts.
A Technology and Construction Court ruling published this week said that an arbitration hearing can now be held into the claims.
Tarmac had argued that Costain was out of time to claim for the extra £6m in more extensive remedial work than that previously agreed by both sides.
A spokesperson for Tarmac said: “In 2014, there was an isolated issue with the cosmetic finish on concrete supplied to Costain for the M1 barrier, which required remedial work that has now been completed.
“This week’s ruling in the Technology and Construction Court granted Tarmac’s application and confirms that the matter may now be resolved through arbitration in accordance with the contract conditions agreed by the parties.”