Union leaders, who represent about 45,000 workers, have been struggling to find a way to avoid the layoffs after the membership last month rejected a labor-savings deal.

Four of the 15 SEBAC unions voted against the deal. A total of 57 percent of the votes were in favor of it, but under the coalition's current bylaws, at least 14 of the 15 unions have to vote to support any changes to the coveted, 20-year health and retirement benefit agreement that's in place until 2017.

Leaders last week rejected a motion to amend the coalition's bylaws to retroactively affect the vote, saying they wanted to respect the wishes of their members who opposed the deal. However, they did adopt a resolution to consider future changes to the bylaws that would govern future labor agreements. Those bylaws were being reviewed this week.

"We spent 60 days negotiating an agreement that was ultimately good for everybody, short-term-wise and long-term-wise, and clearly the process requiring all but one union to vote for it and representing 80 percent of registered membership is unlikely to be met in the future, based on the process," Malloy said.

"If you told most Americans that 57 percent of a group of people voted for something, they'd find it hard to believe that it didn't pass," he added. "So, we need to understand what the path to approval would be and what it would look like."

Matt O'Connor, a SEBAC spokesman, said the unions want to "get back to discussions as soon as possible."

"That being said, we are glad to see there is mutual interest in both sides in finalizing an agreement," he said in a statement. "And leaders are confident that we can, because so much is on the line — from job cuts to service reductions to contracting-out."

Last week, the Democratic-controlled General Assembly granted Malloy expanded budget-cutting authority to balance the two-year, $40.1 billion budget without the $1.6 billion in anticipated savings from the labor savings and concessions agreement. Part of that bill, which passed during a special legislative session, gave the unions until Aug. 31 to pass "an agreement" between the state and SEBAC, not making specific reference to the deal that failed.

The legislation requires Malloy to submit his proposed reductions to the legislature by July 15. Lawmakers would be able to hold a hearing on those cuts and make changes if they decide that's necessary.

But many legislators have been holding out hope that the administration and the unions can somehow reach a deal and make the wide-ranging budget cuts and layoffs moot.

"I think that it's in everyone's best interest to be able to have this agreement in place. I know that it's asking a lot, but it's important," said Rep. Tim O'Brien, D-New Britain. "Most of the state employees voted for this. A lot of people, I think, unfairly attack state employees and the fact that the 57 percent of them voted for it show that they are interested in being part of the solution to the budget crisis."