NEW YORK (CNN/Money) - American Airlines Chief Executive Officer Don Carty apologized Monday for "past communications mistakes" involving controversial compensation plans. But anger over the perks still may prompt labor unions to void a deal that saved the No. 1 airline from bankruptcy.

American Airlines CEO Don Carty apologizes to the employees and to the union leaders over controversial compensation plans and asks for forgiveness. Play video (QuickTime, Real or Windows Media)

Carty, speaking at a news conference at company headquarters in Fort Worth, Texas, said he "failed" in not giving the airline's three main unions "full details" about a series of bonuses and pension guarantees awarded to the brass.

Moments before Carty spoke, a second American union said it wanted a revote on last week's cost concessions that appeared to have saved the world's biggest airline from bankruptcy.

Angered by the awards to executives, the Transport Workers Union, which represents most ground workers, said it would hold a new vote on the labor concessions it made last week.

The TWU joins the Association of Professional Flight Attendants, which also vowed to take a new vote on the contract that American says it needs to avoid Chapter 11.

After making $1.8 billion in cost cuts in last week's votes, the labor unions including the pilots' association said they were duped when they learned about retention bonuses and pension funding for senior management that would come even during bankruptcy. The company Friday rescinded the compensation and Carty Monday repeated his vow to take a 33 percent pay cut this year.

Calling American's finances "precarious," the CEO said Monday that the airline is still on the "precipice of bankruptcy."

The back-and-forth possibility of bankruptcy seemed to gain traction Monday when shares of AMR Corp. (AMR: Research, Estimates), which runs American, fell $1.15 to $3.85.

United Airlines and US Airways filed for bankruptcy protection last year as the industry lost billion of dollars when high labor costs combined with a sharp travel slowdown.