The unions represent more than 50,000 workers including pilots, flight attendants and ground workers at American, the nation's third-largest airline.
American and its parent, AMR Corp., are operating under bankruptcy protection.
Doug Parker, the chairman and CEO of US Airways Group Inc., said Friday that to win a merger with American, his company still needs support of AMR's creditors, management and board of directors.
"But this is obviously an important first step along that path, and we are hopeful we can all work together to make this happen," Parker said in a note to US Airways employees.
Parker said a merger could save about 6,200 jobs at American, or nearly half the jobs that American wants to eliminate. He said he would keep both airlines' current hubs and planes to create a bigger company that could compete against United Airlines and Delta Air Lines.
The three unions at American said in a statement that a merger with US Airways is the best way to fix American, which filed for bankruptcy protection in November.
The unions oppose American's plan to cut 13,000 union jobs and sharply cut labor costs to return to profitability. American is seeking to throw out contracts with the unions that govern pay, benefits and work rules, and impose its own terms on employees.
US Airways and the unions said that they have agreed on terms for collective bargaining agreements if there is a merger between the airlines. However, no such deal exists.
Thomas Horton, CEO of American Airlines' parent company, AMR Corp., has indicated he would prefer that AMR remain independent but is open to a merger after his company emerges from bankruptcy protection. Friday's gambit by US Airways and the American Airlines unions could complicate Horton's strategy, however.
In morning trading, shares of US Airways Group Inc., fell 27 cents, or 2.8 percent, to $9.24.