The Boston Public Library is staring at a $580,000 gap in its budget under expected funding levels from the city and the governor’s budget, library officials said last week.

Options that the library system’s board of trustees may have to consider include closing the central library in Copley Square on Sundays, and closing up to four branches, as they had attempted to do last year.

Mayor Thomas Menino’s administration backed off a proposal to close the branches, including the one in Lower Mills, after community outcry and an agreement by state lawmakers to send $350,000 to keep the libraries open through the rest of the fiscal year, which ends June 30.

For fiscal year 2012, Gov. Deval Patrick has set aside $2.4 million for the library, when $3.9 million is needed to keep all the facilities open, library officials said.

The library system’s overall budget has fallen to $40 million in fiscal year 2011, from $48 million in fiscal year 2009. The budget is fed by city and state funding, as well as trust funds, donations, and some federal funds.

Other options library officials presented to trustees include reducing the number of hours at branches, and reducing additional book purchases for circulation.

“If we don’t get the governor’s funding, all of these options would have to re-examined, in any case,” Boston Public Library president Amy Ryan said at a trustees meeting last week. She stressed that the figures aren’t final and are subject to change. “They could go up, they could go down.”

Closing between one and four branches would save between $250,000 to $1.5 million, while closing the central library on Sundays would save approximately $250,000 from October through May, according to the BPL.

The library has to submit its budget recommendations to the city by the end of March. That comes before the House and Senate are expected to release their budget versions and come to an agreed-upon bill to send to the governor for his signature in the summer.

Menino has written to the chairman of the House budget-writing committee, Rep. Brian Dempsey of Haverhill, requesting the $3.9 million for the 26-library system. That figure would be an increase of $1.1 million from fiscal year 2011, Menino noted, but the request would still come under 50 percent of the fiscal year 2009 levels.

“Statutory funding levels would allow the BPL to maintain its current level of service – keeping all 26 branches and Copley open, maintaining Sunday hours at the Central Library, filling critical vacancies system-wide, and initiating a state-wide digital library and repository,” he wrote in a letter dated Feb. 15.

The letter also outlines his requests for local aid, the passage of a municipal health care bill, and assistance in the city’s efforts to combat homelessness.

State Rep. Byron Rushing, whom Menino appointed to the board of library trustees last year, asked library officials for a longer list of options.

Asked about library closures after the meeting, Rushing said, “I don’t see any reason to do that unless we come up with a plan of better ways we can be servicing neighborhoods.”

Paul La Camera, a fellow trustee, said he was reluctant to consider closing the central library on Sundays when it is open from 1 p.m. to 5 p.m. Officials say the library is usually busy that day and full of families and tourists. Keeping it open on Sunday is “part of what makes us a world class city,” La Camera said.

The public is invited to “budget roundtable” discussions that will be held on Tues., March 15, from 6:30 p.m. to 7:30 p.m. at the Central Library in Copley Square and the next day, March 16, at the Codman Square branch, also from 6:30 to 7:30 p.m.