ST. PAUL — Gov. Mark Dayton announced a new push Tuesday to offer paid leave to new parents among the state’s 35,000-member workforce, a proposal his Democratic allies have deemed a first step toward their goal of extending sick and parental leave to all Minnesota workers.

Dayton is asking the Legislature to dip into a budget surplus this year to fund the new benefit, estimated to cost about $6 million annually. It would mean mothers and fathers with new children who work for the state would qualify for six weeks of paid leave rather than exhaust vacation days or forego paychecks. The benefit would apply to parents of biological or adopted children.

More generous leave policies are a rallying cause among Democrats, but the proposal faces an uncertain path in Minnesota’s Republican-controlled House, where a broader measure gained little steam last year. Dayton said he had not yet discussed the possibility with House leadership, and a top House Republican expressed concern that the governor’s plan would “widen the gap” between public and private-sector workers.

Even as they stood behind the governor’s more modest proposal, top Democrats in the House and Senate said they still hoped to require employers across the state to provide both paid sick and parental leave.