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Union officials met Saturday to continue conversations about another possible strike against five Allina Health hospitals, where as many as 4,800 nurses could walk out in a labor dispute over health benefits, workplace safety and staffing.

The nurses rejected a proposed Allina contract on Thursday, voting against it with a wide enough margin to authorize a strike. They have to give Allina 10 days' notice before a strike can occur.

On Tuesday, the parties will meet to continue negotiations. An open-ended strike may come — for the first time since 1984 — if those talks break down.

Leaders of the Minnesota Nurses Association (MNA), the union representing thousands of nurses, met for more than five hours Saturday to prepare for the discussion. They said they are hopeful talks will be productive and avoid further actions.

"We're playing battleship here," said Rick Fuentes, an MNA ­spokesman. "We don't know where the other side is moving."

An earlier strike by the nurses was limited to seven days in June. Allina's financial report for the first half of 2016 shows it cost the health system $20.4 million to weather the walkout by hiring replacement nurses recruited from around the country.

A longer strike could be much more expensive, but health system leaders said they are prepared for that and plan to hold firm on their demand that the nurses transition to Allina's corporate health plans. Allina has said eliminating four union-backed plans and transferring nurses to their corporate options would save $10 million per year.

Allina later countered with an offer to discontinue only two of the union plans, though nurses would have to bear most of the cost increases to keep the remaining plans afloat — a move executives called a major compromise.

"It was disappointing to read that the focus of the union's meeting today was on planning a strike, rather than working on a solution," Allina spokesman David Kanihan said Saturday. "We look forward to hearing what the union's ideas are when we meet on Tuesday."

Nurses have argued that the union plans with their low-deductible benefits are important, given the high rate of workplace illnesses and injuries in their profession. But they also have sought changes to improve workplace safety and increase nurse-to-patient staffing.

Kanihan said a health insurance agreement with the union would be "absolutely essential to reaching a contract and putting this behind us."

Allina officials say the safety and care of its patients is the organization's first priority and, while they hope to avoid a strike, they have a plan in place should it occur.

Staff writer Jeremy Olson contributed to this report.

Liz Sawyer • 612-673-4648