WASHINGTON (AP) — House Speaker John Boehner predicted Thursday that a majority of House Republicans will end up supporting some kind of compromise to avoid a government default. Democrats insisted that higher tax revenue be part of a deal.

White House budget chief Jacob Lew told reporters at the Capitol that “I’m unaware of a deal” between President Barack Obama and congressional Republicans and he repeated that “we’ve made clear revenues have to be included.”

All sides pushed against media reports that Obama and House Speaker John Boehner, R-Ohio, were near an agreement on a grand bargain trading $3 trillion or so in spending cuts and a promise of $1 trillion in tax revenues through a later overhaul of the tax code as part of a deal to extend the government’s borrowing authority.

“We’re not close to a deal,” White House spokesman Jay Carney said.

“While we are keeping the lines of communication open, there is no ‘deal’ and no progress to report,” Boehner spokesman Michael Steel said.

What seems clear is that the White House and Boehner continue to negotiate in hopes of a large deficit-cutting package as an Aug. 2 deadline looms on extending the debt limit.

“”There has been a meeting of the minds … to try and get a grand bargain, a big deal,” said Rep. Chris Van Hollen, D-Md. “I think both sides have remained focused on that.”

“I only know what you know about the agreement — the potential agreement,” Senate Majority Leader Harry Reid, D-Nev., said. “What I have to say is this: The president always talked about balance. That there had to be some fairness in this. That this can’t be all cuts. There has to be a balance. There has to be some revenue.”

Earlier, Reid called up for Senate consideration a House-passed “cut, cap, and balance” measure that would increase the debt limit by $2.4 trillion, but which would condition the increase on immediate spending cuts and a balanced budget amendment to the Constitution. But he said it “doesn’t have one chance in a million of passing the Senate.”

At a news conference, Boehner told reporters, “Frankly, I think it would be irresponsible on behalf of the Congress and the president not to be looking at back-up strategies for how to solve this problem.”

“At the end of the day, we have a responsibility to act,” the Ohio Republican said.

Asked whether GOP lawmakers supporting the House’s “cut, cap and balance” debt limit measure would be unwilling to ultimately compromise, Boehner said, “I’m sure we’ve got some members who believe that, but I do not believe that would be anywhere close to the majority.”

Boehner’s talk of possible accommodation in the protracted political stalemate over federal budget policy came as the Senate took up the tea party-backed House legislation Thursday.

Democrats argue that the measure would impose untenable spending restraints and set spending levels, as a percentage of the overall economy, on par with the mid-1960s — before the advent of Medicare and automatic Social Security cost-of-living adjustments.

The development Thursday reflected the reality that there’s more talk than progress as official Washington wrangles daily over finding a way out of a debt dilemma that has the government sliding inexorably toward a first-ever default on its financial obligations.

Senate Minority Leader Mitch McConnell, R-Ky., said Thursday the legislation now before his chamber would be an opportunity for lawmakers to “go on record in support of balancing our books or against it.” He urged Democrats to join GOP senators in backing it.

Democrats are expected to kill the measure — which they say would demand debilitating cuts to Medicare — in a vote on Saturday if not before.

Meanwhile, momentum on a separate bipartisan budget plan by the Senate’s “Gang of Six” seemed to ebb as critics warned the measure contains larger tax increases than advertised and it became plain that the measure comes too late and is too controversial to advance quickly — particularly as a part of a debt limit package that already would be teetering on a knife’s edge.

Absent a breakthrough between Obama and Republicans, there is a hotly contested backup plan by McConnell that would give Obama broad new powers to obtain increases in the government’s borrowing unless blocked by veto-proof two-thirds margins in both the House and Senate.

Many conservative Republicans are in an uproar over the McConnell plan, and more than 70 House members signed a letter circulated by members of the conservative Republican Study Committee calling on Boehner to come out in public opposition to the McConnell-Reid plan..

In a shift, said Wednesday that Obama would back a short-term deal to prevent a disastrous financial default on Aug. 2 but only if a larger and still elusive deficit-cutting agreement was essentially in place.

Officially, the president continued to push for a big compromise that would cut the nation’s budget deficit and extend the government’s tapped-out borrowing power above the current $14.3 trillion cap. Obama had threatened to veto any stopgap expansion of the nation’s debt limit, at one point last week even challenging House Majority Leader Eric Cantor, R-Va., not to call his bluff about it.

Carney said if a divided Congress and the White House can agree on a significant deal, Obama would accept a “very short-term extension” of the debt limit to let bigger legislation work its way through Congress.

Obama also is open to the McConnell plan, but it seems barely aloft due to fervent tea party opposition in the House. The hope appears to be that such an option will look a lot better to the House in a week or so, given the lack of other ideas.

The Gang of Six plan has come under assault from critics like House Budget Committee Chairman Paul Ryan, R-Wis., who say the plan would increase taxes by $2 trillion over the next 10 years instead of the $1 trillion-plus claimed by proponents like Conrad — a development likely to stunt momentum among Republicans.

The revenue increase is larger than advertised because the $1.2 trillion in new taxes comes on top of an underlying assumption used by Obama’s deficit commission — and incorporated by the Senate group in its plan — that the Bush-era income tax brackets for family income exceeding $250,000 would revert to the higher, Clinton administration levels. The deficit panel’s assumption was made before Obama buckled in December and signed a full extension of the Bush tax cuts.

Copyright 2011 The Associated Press.