One of Martha Sheridan’s first tasks as Greater Boston’s new tourism chief was to ask lawmakers for a bigger piece of the hotel tax pie.

That didn’t get far: Another budget season, another round of level funding.

So Sheridan returned this month, with a different ask: Let hotels impose their own assessments, on top of existing room taxes, to raise millions for tourism promotion.

This idea might have traction. The concept of tourism taxation districts has kicked around the State House for a few years but hasn’t gone anywhere. Sheridan’s counterpart in Springfield was the original proponent, but that group is smaller and has less clout than the Greater Boston Convention & Visitors Bureau.

The momentum shifted in August, when the Greater Boston council’s board got behind the bill. Mary Kay Wydra, president of the Greater Springfield Convention & Visitors Bureau, calls the endorsement a “game-changer.”

The bill would allow regional tourism districts to impose new room assessments of up to 3 percent. Sheridan doesn’t like to call them taxes, because state government would administer the money but would not decide how to spend it. A majority of hotels within a particular district would have to agree on the surcharge.

The new funds would be controlled by the local tourism council to promote the region. In Greater Boston, Sheridan says, a 1.5 percent assessment could mean $27 million more.

Talk about a game-changer. The state collected $284 million in hotel taxes in the last fiscal year, but only $10 million went for tourism. Regional tourism councils divvied up $6 million, with Greater Boston getting just over $2 million. The rest went to statewide marketing. (A sliver of new casino taxes is also on its way.)

Sheridan, who came here in January from Providence, knows many other cities and states that compete for tourists’ attention get much larger public subsidies.

That’s why some prominent tourism industry leaders, such as Boston Duck Tours chief executive Cindy Brown and Seaport Hotel general manager Jim Carmody, are actively campaigning for this bill. Brown says she has spent years knocking on doors at the State House on behalf of the industry with little to show for it. Carmody says a more reliable funding source is crucial; tourism is usually among the first items to get the ax in midyear budget cuts.

Meanwhile, lawmakers are also considering legislation to expand the Boston Convention & Exhibition Center, a project that would be funded in part through the sale of the Hynes Convention Center. Sheridan’s predecessor, Pat Moscaritolo, says Boston hoteliers were once reluctant to back a regional assessment in part because they also worried hotel taxes might be raised to pay for the convention center expansion. That’s off the table now, based on the revised plan Governor Charlie Baker unveiled last month. Plus, the Back Bay hoteliers will want some relief if the Hynes is slated for closure. Maybe this legislation could help.

Carmody might be sold. But some of his peers in the hotel business apparently are not. Paul Sacco, president of the Massachusetts Lodging Association, says his board will likely discuss the bill next week. Some members love it, he says, some are lukewarm, and some hate it.

In the end, the toughest crowd might be at the State House. Sheridan has learned how hard it can be to convince lawmakers to part with money from an existing revenue source. Now, she will find out whether they can support a new kind of levy over which they have no control.