Barack Obama has warned that the US is "running out of time" to raise the limit on US government borrowing and that failure to do so will lead in effect to a tax increase for all Americans, because a downgrade of the country's credit rating would cause an interest rate rise.

The president's warning was reinforced by a threat from the ratings agency Standard & Poor's to strip the US of its AAA standing if no long-term political deal is reached to tackle government spending and debt.

As Obama and Republican leaders in Congress continued to wrangle over the terms for approving an increase in the US's $14.3 trillion (£8.9tn) debt ceiling by the 2 August deadline – with Republicans rejecting Obama's demand that tax increases for the wealthy accompany sharp budget cuts – the president warned ordinary Americans of the seriousness of the situation.

"This is not some abstract issue. These are obligations that the United States has taken on in the past. The Congress has run up the credit card and we now have an obligation to pay our bills. If we do not it could have a whole set of adverse consequences. We could end up with a situation, for example, where interest rates rise for everybody all throughout the country, effectively a tax increase on everybody," he said.

But Obama also told a White House press conference that while the situation was serious, it could be resolved. "We don't have to do anything radical to solve this problem. Contrary to what some folks say, we're not Greece, we're not Portugal.

"It turns out that our problem is we cut taxes without paying for them over the last decade ... We fought two wars. We didn't pay for them. We had a bad recession that required a recovery act and stimulus spending."

S&P, which follows Moody's in warning of a possible downgrading of the US's top credit rating, put America on negative watch on Thursday and said there was "at least a one-in-two likelihood" that it could downgrade its debt "by one or more notches ... if we conclude that Congress and the administration have not achieved a credible solution to the rising US government debt burden and are not likely to achieve one in the foreseeable future".

Obama said that political leaders "should not even be this close to a deadline on this issue", but he stood firm in his opposition to Republican plans for $2.4tn in immediate spending cuts. The president said to achieve that level of savings without added tax revenues would require the "gutting" of social programmes that he could not support. He said that when ordinary Americans are asked to contribute more to retirement and healthcare programmes, then "millionaires and billionaires can afford to do a bit more".

Republican leaders in the US Senate appeared to be edging closer to an emergency deal with Democrats that would permit the president to raise the debt ceiling unilaterally, but there was continued opposition from fiscal conservatives in the House of Representatives who view such an arrangement as a victory for the White House. Obama said that the Republicans had "boxed themselves in" with election commitments.

The Republican leadership in the House of Representatives said it won control of the lower house of Congress in last November's election with a mandate to sharply cut government spending without any increase in taxes.

The Tea Party movement and fiscal conservatives intend to hold newly elected House members to that commitment, and warn that any deal with the president that does not include deep cuts or permits tax increases will be viewed as a betrayal.

Obama described any temporary solution that did not tackle long-term spending problems as the least attractive option. "We have a unique opportunity to do something big. We have a chance to stabilise America's finances for a decade, for 15 years or 20 years, if we're willing to seize the moment," he said.

John Chambers, chairman of S&P's sovereign ratings committee, also warned that an interim solution of the kind under discussion in the Senate would not be good enough and that Washington must tackle the long-term debt issue. "If you get a small agreement, that will lead to a downgrade," he told Reuters.