The offer did not manage to attract the minimum percentage necessary.
The text approved includes the creation of a new resource fund for pipelines.
Although presidential frontrunner Andrés Manuel López Obrador and the future airport operator both agree that putting the project out to concession could work, they differ on at what stage it should be awarded.
The company alleges inconsistencies in the bidding rules for the project in the capital of Brazil's Ceará state.
Macri and his advisors reportedly urged provincial administrations to finance their own projects using mechanisms such as PPPs.
Although the publicly-run fund will be the project's concessionaire, a tender is expected to be launched for the operation and maintenance of the line.