Negotiators for the Allied Pilots Association say they hope to present a deal to the union's board of directors this week.
American and parent Fort Worth-based AMR Corp. have been operating under bankruptcy protection since last November.
A message Sunday to union members says negotiators recently finalized contract language to the point that there are only a handful of open paragraphs.
The issues include pay raises, furlough protection and limits on American's ability to use smaller planes typically operated by regional airlines such as American Eagle and SkyWest.
American's eight other union labor groups have approved cost-cutting deals.