Legislative leaders and Democratic Gov. Dannel P. Malloy will decide Tuesday whether to convene a special session later this week to approve a newly restructured hospital tax before Oct. 1.

If they don’t they risk leaving over $343 million in federal Medicaid reimbursements on the table.

Under a complicated structure that’s been endorsed by the Connecticut Hospital Association, the state has decided to increase the amount of money it taxes hospitals in order to extract more money from the federal government. As part of the deal, the state has agreed to return more money to the hospitals, which is why the organization representing them has agreed to the deal.

The hospital tax increase was part of both the Democratic and the Republican budget proposal. It would increase the tax from $556 million to about $900 million, but it would double the amount of money hospitals receive back from the state.

“We appreciate that legislators recognize the importance of moving forward the hospitals’ agreement with the Administration,” Jennifer Jackson, CEO of the Connecticut Hospital Association, said Monday. “By moving quickly to make the federal deadline at the end of the month, we’ll be able to maximize federal funding, which will help patients, communities, our hospitals, and the state.”

It wasn’t clear Monday whether Connecticut could secure the increased reimbursement and make sure the rate hike was retroactive to July 1 if they didn’t get the request to the federal government by Sept. 30.

“We are hopeful that we can preserve the potential federal match, but until we have worked through all the federal compliance issues we cannot know for sure how successful we will be,” Secretary of the Office of Policy and Management Ben Barnes said Monday.

Adam Joseph, a spokesman for the Senate Democratic caucus, said it would be a topic of conversation for legislative leaders to discuss with Malloy on Tuesday.

While the hospitals weren’t concerned about it being separate and apart from the rest of the budget discussion, the Malloy administration seemed to want a more comprehensive approach.

“The state needs to remain focused on getting a comprehensive biennial budget, and we should be cautious about piecemeal efforts that would undermine or distract from that important goal,” Chris McClure, a spokesman for Malloy, said. “Nevertheless, we continue to work with hospitals to find a mutually beneficial long-term arrangement. Such an arrangement would help us get to a budget agreement and could provide long-term predictability for both state government and one of our most important industries. If we can get to such an arrangement by October 1 we will all be better off.”