NEW YORK (CNN/Money) - Shares of United Airlines' parent UAL Corp. jumped sharply on Thursday amid hopes that new concessions offered by its unions would be enough to keep the carrier out of bankruptcy court.

UAL (UAL: up $0.73 to $2.94, Research, Estimates) stock finished 33 percent higher following the unions' offer late Wednesday to save the company $5 billion in labor costs over the next five years. That fell short of the $9 billion in savings over six years that the company was seeking, however.

The airline said it will negotiate with each of five unions individually how to divide up the $1 billion per year that a union coalition offered.

"We welcome the framework from the union coalition, which has worked diligently to forge constructive recommendations for addressing some of United's most immediate financial needs," said management's statement.

The unions said that they believe that other unspecified cost saving initiatives could help to improve United profitability by $2 billion-to-$3 billion a year.

"United has the most committed employees and strongest airline franchise in the world. We will not let it fail. We look forward to working with you to implement the coalition framework," said a letter from union leaders to UAL CEO Glenn Tilton.

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The company had earlier warned that without concessions from labor and federal loan guarantees, it would be forced to file for bankruptcy court protection some time in the fourth quarter.

The union offer itself is lacking many key details, including the allocation of the savings among the unions and specific cost saving measures to be taken.

But the agreement on the framework and the proposed savings level is seen as significant, even though initiatives have not yet been determined and will be the subject of further discussions with United Airlines.