HARTFORD — Connecticut state legislators watched the seasons change as they've grappled with how to balance a budget that's beset by a multi-billion-dollar deficit.

But as the temperatures have grown cooler, there's now some hope that lawmakers could be finally warming to the notion of reaching a bipartisan agreement.

For the past week, Democratic and Republican legislative leaders have been holed up in the state Capitol, without Democratic Gov. Dannel P. Malloy, combing line-by-line through budget documents. They said they have been discussing ways to not only cover a projected $3.5 billion deficit in a roughly $40 billion two-year budget, but to make lasting fiscal changes in hopes of stopping what's become a cycle of budget crises in one of the nation's wealthiest states.

"I think what we've done over the last few days has been a really good step forward and I think we're moving in the right direction," said Democratic House Speaker Joe Aresimowicz, of Berlin.

"Getting there is tough, but we're taking a step forward each and every day," he added, without committing to when or if a final deal may be reached.

This all comes as lawmakers face mounting pressure from municipalities, schools and nonprofit social service agencies to reach a deal. Connecticut has been without a budget since the fiscal year began July 1 and that has led to program cuts and layoffs.

A look at Connecticut's confounding budget situation:

Q: Why is the state in this mess?

A: Sagging state revenues, slow economic and job growth, a court ruling that determined Connecticut's education funding system is unconstitutional and a General Assembly with the closest partisan make-up in recent history are among the host of reasons why the state still does not have a budget in place. Since Malloy kicked off the budget process in February, the projected deficit has worsened. It grew from $1.7 billion in the first year of the two-year budget to $2.3 billion, and to $2.7 billion in the second year. While the combined projected deficit of $5 billion has since dropped to $3.5 billion after a labor concessions agreement with state unions, it has still posed a difficult challenge.

Q: What's happening without a state budget in place?

A: Malloy has been forced to use his limited executive spending authority to run state government since July. His executive order, however, has reduced funding for social service programs, such as day services to people with developmental disabilities and initiatives serving people leaving prison. This month, cities and towns are bracing for $140 million in school funding cuts — the first of a total of $557 million. The state's largest teachers' union, the Connecticut Education Association, on Wednesday joined several towns in seeking a court order to stop the reductions, arguing there will be widespread layoffs and program cuts.

Q: What's next?

A: Even if legislators reach a tentative budget agreement, they still hope to get the governor's seal of approval. He already vetoed a Republican-crafted budget that passed the General Assembly with a handful of Democratic votes. There wasn't enough support to override Malloy's veto at that time. The governor, who expressed frustration Thursday over the slowness of the process, has warned that he's willing to veto another budget, even a bipartisan one, if he believes it includes "gimmicks" to cover the red ink, such as reducing payments to state employee pension funds.

"I think that people would really be making a mistake if they totally ignored my veto message," he said, adding how he plans to release a newly revised version of his own budget for lawmakers to consider.