In the discussions about the “fiscal cliff” consuming Washington, a powerful new group stands out, touting the virtue of bipartisanship and shared sacrifice.

The business leaders who set up the Campaign to Fix the Debt appear nearly every day on network talk shows and have won coveted time with President Obama in pushing for increased tax revenue, reduced government spending, and changes to Social Security and Medicare. The group’s leaders met Wednesday with lawmakers on Capitol Hill and returned, yet again, to the White House.

But the organization has an ongoing problem: convincing skeptics that it represents more than the self-interest of its chief executive members, whose companies have provided nearly all of the group’s $43 million budget.

Several top corporate executives have been pressing for changes to the tax code that would cut the taxes multinational companies pay on overseas profits. Dave Cote, the chief executive of Honeywell and a leading figure in Fix the Debt, has been a prominent advocate of this tax change.

At the same time, companies represented in Fix the Debt also belong to a coalition that has sought to cut corporate tax rates, even though many of the companies pay low effective rates.

The corporate tax agenda has been criticized by some within the business community, especially small firms and partnerships that pay taxes based on individual income tax rather than corporate rates. They fear that the chief executives in Fix the Debt may be comfortable with a rise in individual income tax rates, knowing that it could create flexibility for cutting corporate taxes during tax-reform deliberations next year.

Fix the Debt officials say they back broad principles espoused by Erskine Bowles, who was President Bill Clinton’s chief of staff, and former senator Alan Simpson (R-Wyo.), who were on hand for Wednesday’s events. In 2010, the men led a commission to reduce the debt, drawing bipartisan support.

Still, liberal as well as conservative groups are circulating reports alleging that the chief executives who make up Fix the Debt run companies that have been promoting a “rash of corporate tax breaks” in the name of pro-growth tax reform. Beyond that, the group’s chief executive members have been criticized by the left-leaning Institute for Policy Studies for seeking cuts in Social Security and Medicare “while sitting on an average of $9 million each in retirement funds.”

Fix the Debt has said it is not backing any specific plans for raising tax revenue or cutting government spending. Still, the chief executives received a warning Wednesday from House Speaker John A. Boehner (R-Ohio).

“One thing Republicans won’t be party to is a deal that protects big businesses and preserves special-interest tax breaks while raising tax rates on the small businesses,” Boehner said in a statement issued after he met with the business leaders. A spokesman for Fix the Debt said the organization received a warm reception at every event Wednesday and described Boehner’s comment as consistent with the organization’s approach.

“The Campaign to Fix the Debt recognizes that getting consensus on these issues requires compromise, and our supporters are prepared to do their part,” said the group’s spokesman, Jon Romano.

The group’s leaders say their goal is to help provide political support and cover for lawmakers who show the courage to tackle such challenges as changing the tax code and the Social Security and Medicare systems.

“We’re not trying to hurt anyone. Rather, we’re advocating a shared American solution to an American problem,” Cote said. “This will require everyone’s participation in some way. This is not about pushing the burden on any group to the exclusion of others.”

Fix the Debt is deploying sophisticated political tactics to boost its image and win supporters. The group has been building a grass-roots network and has formed chapters in 17 states, with more due to open soon. The number of corporate members has jumped to 130 from 80 in late October.

In addition to Cote, the group’s chief executives include such corporate leaders as Lloyd Blankfein of Goldman Sachs and Mark Bertolini of Aetna. The three were among chief executives and others in Washington on Wednesday for Fix the Debt events on Capitol Hill, which included private meetings with the leadership. Blankfein and other CEOs later went to the White House, where they met with President Obama and other officials.

But even as the chief executives — accompanied by Simpson and Bowles — made their rounds, they were preceded and followed by labor advocates offering an opposing view, particularly on changes to entitlement programs.

Fix the Debt is classified as a 501(c) nonprofit group, meaning it does not have to release the identity of most of its donors. The president of the organization, Maya MacGuineas, has sought meetings with both liberals and conservatives on an almost daily basis. But Fix the Debt has made limited headway with prominent Democrats and their allies, the open door at the White House notwithstanding.

The challenges the group faces were evident when Washington civil rights leaders gathered at one of several Fix the Debt breakfast meetings this month. Making the case on behalf of Fix the Debt were two well-regarded Democrats affiliated with the group: Former congressman Vic Fazio of California, now a lobbyist, and Clinton’s former budget director, Alice Rivlin. Although they were welcomed by the civil rights leaders, their pitch for Fix the Debt fell flat.

“I don’t see the civil rights and labor communities signing on,” said Nancy Zirkin, executive vice president of the Leadership Conference on Civil and Human Rights, who attended. “People care far more about creating jobs than reducing the debt.”