Gov. Dannel P. Malloy has enlisted the help of lawmakers on both sides to help him cut $242 million from this year’s budget. He released the framework Friday for how he hopes to get there.

Earlier this month state Comptroller Kevin Lembo projected that the state would end the year with a $415 million deficit. But Malloy’s budget office remains more optimistic, believing it’s $365 million. Malloy used his authority to cut $123 million, but will need the legislature’s help to cut another $242 million to close the hole.

Based on the framework released Friday, it almost looks as if Malloy is proposing $390.5 million in cuts, but that’s before the loss in federal revenue is taken into account. If all the cuts to state agencies are made, the state also will lose about $58.2 million, mostly in the Department of Social Services.

The budget framework also seeks to recover about $22 million in additional tax revenue on electric generators and corporations. The budget framework says that state is missing out on $10 million it should be receiving from electric generation companies because of a drafting error.

The generation tax levied for the first time last year,  is set to expire in July 2013, but Ben Barnes, Malloy’s budget director, said he’s not thinking past the end of the year. He said it was a drafting error that allowed some companies to find a loophole that the administration wants to close.

The administration is also looking at converting a $10-million stem cell program currently funded with tobacco settlement funds. The state will take the money for the general fund and put the stem cell program on the state’s credit card instead.

In a statement, the governor said the final target number could change and will be determined later by an agreement between his administration and the Office of Fiscal Analysis.

“It is my hope — and I’ve communicated this to legislative leaders from both parties — that the final plan will be a bipartisan one,” Malloy said. “That’s my goal, and if all sides come to the table in good faith, I’m hopeful we can get it done.”

The governor also issued a call for a special legislative session and scheduled a meeting for Monday, where his administration and representatives of all four legislative caucuses will meet to consider specific line items for cutting. He urged everyone to come to the meeting with an “open mind.”

“We are very mindful of the fact that most of the cuts that will end up being part of the final package will have a real impact on people’s lives, so we want to do this as carefully as possible. But the reality is we have no other choice; spending must be cut,” Malloy said.

The “roadmap” Malloy released Friday adds to the $123 million in cuts he made last month under his own rescission authority.

Malloy is recommending that appropriations to the Department of Social Services be reduced by more than $121 million. He had previously cut $32 million from the agency. That’s a total of 2.65 percent of the department’s budget.

It seems as if lawmakers from both parties are willing to work with Malloy on closing the deficit.

Republican leaders from the House and Senate met with Malloy for about 25 minutes Friday morning to discuss the special session. House Minority Leader Lawrence Cafero said it was a constructive meeting.

At one point during the closed-door meeting laughter could be heard from out in the hallway. Following the meeting, Malloy said his senior communication adviser, Roy Occhiogrosso, had made a joke.

But Cafero said there is tough work to be done in the days ahead.

“Things are going to happen rather quickly. We’ll probably be together as a whole group all of next week and frankly we’re not going to stop until we solve this,” he said.

Senate Minority Leader John McKinney said both the governor and Republicans have a genuine desire to close the budget deficit in a bipartisan manner.

“I think we’ve come together on a process that can make that happen and it’s going to be a good process,” he said.

Asked whether there was any resistance among Republicans to helping with potentially unpopular cuts to balance a budget crafted by Democrats, Cafero said, “No.”

“They know it’s not going to be easy. There’s these stereotypes that we’ve often called for cutting, but no one has ever said we relish the opportunity,” he said. “I think my caucus knows there’s a tough road ahead.”

McKinney said Republicans ran for office to help make the state a better place and believe they have good ideas to get there.

“There’s the added benefit of demonstrating to the people of Connecticut that we can actually get our business done in a bipartisan manner, which has not always been the case,” he said.

Cafero said most Connecticut residents aren’t keeping a partisan scorecard and just want to see the state’s problems solved.

“As John said, they elect each individual to go to Hartford and represent them. We’re faced with a problem it’s our job to represent them. We happen to be in the minority party but we represent a third of the men, women, and children who live in this state and I think they want to know they also have a voice at the table and the governor’s facilitated that process,” he said.