Hartford — Gov. Dannel P. Malloy’s budget chief offered the state’s regional tourism districts a glimmer of hope Friday, suggesting budget resources could be juggled to provide the districts with some support.

Ben Barnes, secretary of the Office of Policy and Management, told members of the legislature’s tourism caucus that he would not rule out a plan that would keep the three districts afloat.

“I don’t know what that arrangement is, but I’m willing to listen,” Barnes said.

More than $1.1 million in funding for the three districts, including the 42-town Eastern Regional Tourism District, was completely eliminated June 30, the day before the current fiscal year began.

Across-the-board budget cuts totaled $130 million, reducing the state’s 2016-17 spending plan to $19.8 billion.

Barnes said he was aware the cuts in regional tourism funding could force the district offices to close.

“I would hope that they can find a way to provide some level of service … through alternative sources of funding,” he said.

Outside a meeting room in the Legislative Office Building, Barnes said there may be some way to reallocate budget resources to “ease the transition” for the regional districts. He said, however, that he would not support a major restoration of their funds.

“If there were another $10 million available, we’d want to put it into statewide marketing,” he said.

The 2016-17 budget still includes $6.4 million for statewide tourism, which primarily involves the “Still Revolutionary” promotional campaign.

Ed Dombroskas, executive director of the Eastern Regional Tourism District, said he was “buoyed” by Barnes’ comments.

“What I heard him say was that the decision to eliminate funding wasn’t final,” Dombroskas said after the caucus meeting. “They’re willing to be flexible.”

Other regional officials were less magnanimous.

“It’s a critical situation,” Dan Bolognani, chairman of the board of directors of the Western Connecticut Convention and Visitors Bureau, said. “We’re on the verge of closing. Lay-off notices have gone out.”

The three regional districts employ fewer than a dozen people.

Jeff Muthersbaugh, vice chairman of the Central Regional Tourism District’s board, said the state’s longstanding interest in centralizing tourism promotion was evident.

“This is fundamental,” he said. “If we don’t have regional districts, we’ll be the only state in the country that doesn’t.”

Muthersbaugh, who owns the Nehemiah Brainerd House Bed & Breakfast in Haddam, said his business couldn’t survive without the promotional services his regional district provides. He suggested lawmakers look to the money generated by the state’s highest-in-the-nation hotel occupancy tax as a source of tourism funding.

Caucus members, including Rep. Diana Urban, D-North Stonington, said the complete elimination of funding for the districts was a bad idea.

Without any state support, Urban said, the districts will find it difficult to tap private funding sources and develop public-private partnerships. Providing some level of funding could serve to “nudge” the districts in the right direction in those areas, she said.

Rep. John Scott, R-Groton, said the savings associated with the elimination of the regional districts would be more than offset by the resulting harm done to local businesses.

“Reducing the marketing budget … will mean significantly less tourist dollars being spent here, less taxes will be collected on lodging and goods, and we will be forced to make further cuts to programs and services to make up for the lost income,” he said.

Sen. Paul Formica, R-East Lyme, who co-chairs the tourism caucus along with Urban, said he was grateful that Barnes, the budget chief, was “willing to be open-minded.” He said the caucus will meet again in the next week or two to continue addressing the situation.

“Quickly,” Dombroskas said. “We’re facing imminent demise.”

b.hallenbeck@theday.com