Pressure is building on Democratic and Republican leaders to end the 28-day shutdown after the nation’s largest federal workers union called for an end to the stalemate as rank-and-rile GOP lawmakers sound the alarm over rising health insurance premiums.

The blunt message from Everett Kelley, the president of the American Federation of Government Employees, is one of the biggest developments in the monthlong standoff.

It appeared to be aimed squarely at Senate Democrats, who have voted a dozen times to block a clean seven-week continuing resolution passed last month by the House, and it came from someone representing 820,000 federal and D.C. workers.

In calling on Congress to “pass a clean continuing resolution and end this shutdown today,” Kelley argued that “both political parties have made their point” and that it’s time to bring hundreds of thousands of government employees back to work.

Senate Democratic Whip Dick Durbin (Ill.) said the statement would have a “lot of impact” and that Democrats would discuss it in the days ahead.

“They’ve been our friends, and we’ve worked with them over the years. I’ve had a good connection with them and I’ve talked to them. They’re in a terrible mess. So many of their workers are not being paid,” Durbin said of the union.

Even so, other major unions, including the American Federation of State, County and Municipal Employees, which represents between 1.4 million and 1.6 million state and local workers, haven’t broken with the Democratic position.

Durbin, the No. 2-ranking Senate Democratic leader, said the expiration of Supplemental Nutrition Assistance Program (SNAP) funding starting Nov. 1 is another major concern.

“The SNAP program feeds 1 out of 8 Americans,........

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