Washington ? After weeks of turmoil, House Democrats reached a shaky peace with the party’s rebellious rank-and-file conservatives Wednesday to clear the way for a vote in September on sweeping health care legislation.

Bipartisan Senate negotiators reported progress, too, on a bill said to extend coverage to 95 percent of all Americans without raising federal deficits.

“We’re on the edge, we’re almost there,” said Sen. Charles Grassley of Iowa, the senior Republican involved in the secretive talks, although a fellow GOP participant, Sen. Mike Enzi of Wyoming, dissented strongly.

Sen. Max Baucus, D-Mont., chairman of the Finance Committee, said preliminary estimates from congressional budget experts showed the cost of the emerging Senate plan was below $900 billion and would result in an increase in employer-sponsored insurance — conclusions that may reassure critics who fear a bloated bill that prompts businesses to abandon the coverage they currently provide.

Across the Capitol, House Democratic leaders gave in — at least temporarily — to numerous demands from rank-and-file rebels, so-called Blue Dogs from the conservative wing of the party who had been blocking the bill’s passage in the last of three committees.

The House changes, which drew immediate opposition from liberal lawmakers, would reduce the federal subsidies designed to help lower-income families afford insurance, exempt additional businesses from a requirement to offer insurance to their workers and change the terms of a government insurance option.

At their core, both the House bill and the plan under negotiation in the Senate are designed to meet President Barack Obama’s goals of spreading health coverage to millions who now lack it, while slowing the skyrocketing growth in health care costs nationally.

Obama has placed the issue atop his domestic agenda, and as recently as two weeks ago was pressing the House and Senate insistently to pass separate bills by the end of July or early August.

The White House issued a statement praising the development in the House, and with appearances in North Carolina and Virginia, the president sought to minimize the significance of the slippage in his timetable.

“We did give them a deadline, and sort of we missed that deadline. But that’s OK,” Obama said. “We don’t want to just do it quickly, we want to do it right.”

In his appearances, Obama stressed that any legislation he signs will include numerous consumer protections, including a ban on insurance company denials of coverage based on pre-existing medical conditions. A White House fact sheet left room for insurers to continue charging higher premiums based on prior health problems.

Rep. Mike Ross of Arkansas, a leader of conservative and moderate “Blue Dog” Democrats, said the changes agreed to by the leadership in the House bill would reduce costs in the bill by about $100 billion over 10 years. But the new break for small businesses, among other changes, also increased costs substantially, so it wasn’t clear that the deal actually generated net savings.

As word of the agreement spread, liberals fired back. “We do not support this,” said Rep. Lynn Woolsey, D-Calif., co-chair of the Progressive Caucus. “I think they have no idea how many people are against this. They can’t possibly be taking us seriously if they’re going to bring this forward.”

Plans to convene the Energy and Commerce Committee for a vote slipped until today as leaders sought to allay concerns of liberals.

“We just need to get everybody on board,” said Rep. Frank Pallone, D-N.J., who chairs the panel’s subcommittee on health.