Five teams are seeking approval to bid a third link, estimated at $500 million, of an $8.7-billion rail line now under way between New Jersey and Manhattan.

The contract, to be awarded next fall by project owners New Jersey Transit and the Port Authority of New York and New Jersey, will involve building 14,600 feet of soft-ground tunnels that run 110 feet under the Hudson River. The project is set to be the nation’s largest public works transit job.