House Speaker Chris Donovan and incoming Speaker Brendan Sharkey emailed their members Thursday to let them know they will probably have to return to the Capitol the week before Christmas to erase an estimated $365 million budget deficit.

“Assuming the deficit certified by the Comptroller in early December is 1 percent or greater of the total general fund appropriations, the governor will have 30 days to submit a Deficit Mitigation Plan,” Donovan and Sharkey wrote in an email to their members.

Prior to the election, the General Assembly had been expected to remain in recess until the full session starts Jan. 9, but recent reports from Gov. Dannel P. Malloy’s Office of Policy and Management and the legislature’s Office of Fiscal Analysis show the deficit increasing at a rate that requires them to act.

Budget Director Ben Barnes told agency commissioners Thursday that the state faced a $365 million deficit this fiscal year and upward of a $2.13 billion deficit over the next two fiscal years. But the only issue that will have to be resolved in December with the current legislature is the $365 million deficit.

“We currently have little information about the governor’s Deficit Mitigation proposals, but as more information is provided to us by the governor’s office, we will review it and pass it along to all of you,” Donovan and Sharkey wrote.

Malloy has the authority to cut 5 percent of any appropriation and 3 percent of any fund in a financial crisis without legislative approval, but to make the $365 million in cuts it’s likely he will need legislative approval. Asked whether the $365 million target is within reach under his authority to cut specific appropriations, Malloy administration officials opted not to commit to an answer.

Republican lawmakers called upon Malloy to issue a deficit mitigation plan last week when they saw the revenue data. They also accused the Malloy administration of being less than transparent about the current fiscal situation.

On Thursday, the governor said information was released as it became available and pointed out that the fiscal year doesn’t end until July, making the current budgetary gap a shortfall rather than a deficit.

“It’s not going to be a deficit. It’s going to be addressed,” he said.

On Wednesday, Barnes said it was too soon to say what would be on the table for potential cuts.

“Government spending. State government spending. That’s about as specific as I’m going to get,” he said. “… You act as if I have in my mind a full list of what we’re going to do and I don’t yet. We’re working hard to do that. Until we get through that process and work with the governor and make sure we’re right on what the deficit is going to be, it’s way premature for me to discuss what any kind of plan in the future’s going to be.”

Meanwhile, the three other caucuses have already notified their members about a potential session.

“Senate leadership has spoken with our caucus members, and they understand there is a possibility that they will be coming in to special session before the end of the year,” Democratic State Senate Deputy Communications Director Lawrence Cook said.

No specific date for the special session has been set, but it’s expected to be held the week of Dec. 17.

Hugh McQuaid contributed to this report.