Hundreds of thousands of New Jersey commuters may be caught up in the state’s first transit strike in more than 40 years as soon as Friday morning.

The engineers who drive New Jersey Transit’s commuter trains have threatened to walk out after midnight Thursday if the union that represents them cannot reach an agreement with the agency on a contract that has been under negotiation for many months.

The engineers have said they are seeking parity in pay with engineers at other commuter railroads in the area. The transit agency’s chief executive, Kris Kolluri, has repeatedly said that it cannot afford the pay increases the union is demanding.

The two sides met with the National Mediation Board in Washington on Monday in a last-ditch effort to resolve their differences. Mr. Kolluri said the discussion was “constructive” and that he looked forward to continuing negotiations in good faith.

In the meantime, New Jersey Transit has drawn up a contingency plan. But Mr. Kolluri said that the additional buses it would offer as a substitute in the event of a strike could handle only about 20 percent of the daily train riders.

The agency has asked commuters to make plans to work from home if their presence in the workplace is not essential. Mr. Kolluri has said a strike could last for weeks.