Brigham and Women’s Hospital and the union representing 3,300 of its nurses failed to reach a contract agreement after more than 10 hours of bitter negotiations Friday, but agreed to talk again next week in another attempt to find a compromise that could avoid a disruptive labor strike.

Without a contract deal, the Massachusetts Nurses Association has vowed to hold a strike on June 27, to protest what it calls the hospital’s disrespect for nurses during ongoing contract negotiations. Brigham is owned by Partners HealthCare, the state’s largest nonprofit health care system.

Negotiators from both sides met at 9 a.m. Friday at City Hall, a venue offered by Mayor Martin J. Walsh, and talked into the evening. They agreed to meet again on Monday.

But the rift between the hospital and the union only appears to be growing. Both sides issued sharply worded statements Friday night.

“We are disappointed that the MNA rejected all of the proposals that the hospital put forward today and refused to continue negotiations into the evening or over the weekend,’’ Brigham’s chief operating officer, Dr. Ron M. Walls, said in a statement.

The union on Friday accused hospital leaders of “doubling down on destroying the Brigham way of excellence in patient care.’’ It said Brigham wanted to maintain “dangerous staffing levels’’ for certain patients and refused to agree to what it called “fair’’ wages and benefits for nurses.

In response, the hospital said it has “made multiple proposals responsive to the MNA’s concerns. The MNA has rejected every attempt at a resolution.’’ It insists it has safe staffing in all its units.

If it happens, a strike threatens to disrupt patient care at one of the city’s busiest hospitals. Brigham has contracted with a staffing agency to hire 700 temporary replacement nurses in case of a walkout.

The union is planning a one-day strike, but hospital leaders plan to lock out the union nurses for five days and use temporary workers for “continuity of patient care and for economic reasons.’’

The hospital wired money to the temporary nursing agency Friday night, per their contract, to make sure the replacement nurses would be ready to work on the day of the planned strike.

If the strike occurs, all parts of the 793-bed hospital would remain open, but many appointments and procedures would be canceled, and many patients may be transferred to other hospitals.

Friday’s negotiating session was the 20th all-day meeting between the two sides. Over nine months of contract talks, they have failed to find agreement on wages, benefits, and other issues.

Brigham nurses receive 5 percent annual raises for their first 18 years on the job. The hospital proposed giving most nurses an additional 1 percent increase over three years. Nurses already at the top of the pay scale would have received a 1.5 percent increase, plus a $500 bonus.

The union sought a 4 percent raise for all nurses over 18 months, on top of the existing step raises. A Brigham nurse, on average, earns $106,000 a year, plus benefits.

Priyanka Dayal McCluskey can be reached at priyanka.mccluskey@globe.com. Follow her on Twitter @priyanka_dayal.