The impasse over the nation's debt ceiling appeared to harden Sunday as leading lawmakers shot down each other's proposals and struggled to explain how exactly Washington will avert a potential fiscal crisis in the next 16 days.

Despite an insistence by members of both parties that the nation will not default on its obligations, they shared little common ground on the way forward following a marathon week of talks hosted by the White House. President Obama's budget director Jack Lew said negotiators continue to talk behind the scenes since the last meeting Thursday, but lawmakers' remarks Sunday betrayed a deepening distrust between the parties on their respective commitment to a deficit-reduction deal.

"This is Lucy and Charlie Brown and a football. And the American people say we're not falling for that game anymore," Rep. Jim Jordan, R-Ohio, said on "Fox News Sunday," expressing concern that Democrats want an agreement to raise taxes now with a "promise" to cut spending later.

Sen. Dick Durbin, D-Ill., likewise bashed a GOP plan to push for a balanced-budget amendment in exchange for raising the $14.3 trillion debt ceiling.

"The bottom line is those who want to push a balanced-budget amendment are saying 'I can't promise you that I won't steal again, but I will vote for the Ten Commandments,'" he said on NBC's "Meet the Press."

More On This...

With officials moving forward on what Lew described as "multiple tracks" to find a way to raise the debt cap, the very-public discord complicates the prospects for any of those proposals.

On one track, House Republicans this week will consider their plan to cut and cap federal spending, while requiring Congress to approve the balanced-budget amendment in order to clear the debt-ceiling increase. But several Democrats argued Sunday that the plan doesn't stand a chance in the Democrat-dominated Senate.

"It does not have the votes in the Senate," Durbin said.

On "Fox News Sunday," Rep. Chris Van Hollen, D-Md., said that plan is "not going to become law," describing the approach as "dangerous."

"What they're proposing is to manipulate the Constitution and use it to impose the Republican budget plan," he said.

On a separate track, Senate Republican Leader Mitch McConnell is working with Democrats on a backup plan that would make it easier for Obama to win a debt-limit increase by effectively requiring a two-thirds majority in Congress to stop him. Accompanying that could be a plan to cut spending by $1.5 trillion and empanel a new commission to examine further deficit reduction.

But Republicans in the House and Senate who support the balanced-budget amendment assailed that approach and said they would not support it. In the Senate, Oklahoma's Tom Coburn and South Carolina's Lindsey Graham were among the Republicans who said they would oppose or were likely to oppose that idea.

In the House, Republican Study Committee Chairman Jordan said conservatives in his chamber as well as the Senate would fight the McConnell backup proposal.

"The McConnell (plan) doesn't have 218 Republican votes. No way. There's all kinds of conservatives in the Republican Study Committee in the House conference who will not support the plan," he said. "The American people sent us here to make big tough choices. They didn't send us here to set up a commission, give the president veto power."

On a third track, Lew said the president is still pushing for "something big" out of both parties.

That effort, which included daily sessions last week with White House and congressional negotiators, has failed to yield a compromise as Republicans balked at Democratic calls for tax increases. Rank-and-file Democrats likewise pushed back on White House and Republican calls to deal with entitlements.

While Lew said a "balanced package" would have to include taxes, Senate Republican Whip Jon Kyl, R-Ariz., said Sunday that Obama's "absolute obsession with raising taxes" will not win Republicans to his side.

Instead, he said the Senate will continue to proceed with the McConnell backup plan barring a change in posture by the president.

On still another track, Coburn said he plans to pitch a proposal that can yield $9 trillion in savings over the next decade. But, speaking on CBS' "Face the Nation," he said "I wouldn't expect it to pass."

The competing proposals and unsteady nature of the talks underscore the precarious position U.S. creditworthiness is in as lawmakers head into the final stretch ahead of what the administration claims is an Aug. 2 deadline to raise the debt cap or face default.

Top credit rating agencies warned over the past week that the country may run the risk of a credit downgrade -- pressuring lawmakers to not only raise the debt ceiling but make a serious effort to tackle the deficit.

Despite the steep challenge, officials expressed confidence they would work out their differences in the end and avoid a default.

Coburn said "there is time, without a doubt" for an agreement.

"There's still time to get something big done," Lew said on "Meet the Press."