The lower house sent the project to the executive branch for the final nod.
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 contract is to widen a stretch of the Pan-American highway from three to six lanes.
US firm Pimco will subscribe to the US$540mn issue, according to a newspaper report.
President Vizcarra's administration is looking to pave over 4,000km of roads by the time it steps down in mid-2021.
The company alleges inconsistencies in the bidding rules for the project in the capital of Brazil's Ceará state.