"It is time to deal with America's problems," he said. "How can you raise the debt limit and do nothing about the underlying problem?"
Obama and Democratic leaders in Congress insist that such congressional responsibilities -- to keep the government running and able to pay its debts -- must be free of partisan political pressure to avoid the kind of collateral damage happening in the current stalemate.
They want what are known as "clean" measures to fund the government for a short period and increase the debt limit, with no accompanying provisions involving contentious deficit reduction measures or GOP efforts to weaken Obamacare.
Once such measures are passed, they say, negotiations can take place on a full budget for fiscal year 2014 that began on October 1 and other issues such as reducing spending on entitlement programs like Medicare, Medicaid and Social Security.
Last week, a House Republican said on condition of not being identified that Boehner told GOP colleagues in private meetings he would not allow a government default to occur. But Boehner sounded more combative on Sunday, saying Obama and Senate Democrats refused to negotiate on either a spending plan to end the shutdown or the debt ceiling.
Senate Democrats are expected this week to take up a debt ceiling bill that would not propose any policy changes or spending cuts demanded by Republicans, according to a Senate Democratic leadership aide.
The aide said Senate Majority Leader Harry Reid could introduce a "clean" bill as early as Monday that could bring a first key procedural vote on Friday.
On the shutdown, Boehner insisted Obama and Democrats were wrong in saying a "clean" short-term spending plan to reopen the government would pass in the House with support from some Republicans and most Democrats.
"There are not the votes in the House to pass a clean CR," Boehner said.
Obama rejected Boehner's contention on Monday, saying the speaker "should prove it" by holding the vote.
"My very strong suspicion is there are enough votes there," Obama said, adding that Boehner "apparently doesn't want to see the government shutdown end ... unless he's able to extract concessions that don't have anything to do with the budget."
Senate Majority Leader Harry Reid suggested the measure would pass the House, and that Americans would realize the government was shut down "for no apparent reason." Both Obama and Reid said Democrats were open to negotiate "anything" -- with the president specifically mentioning health care -- once the government shutdown ends and the debt ceiling gets increased.
House Republicans, however, fear losing their leverage in any talks by giving up those two points without any concessions.
In a new national poll released Monday, most respondents said the government shutdown was causing a crisis or major problems for the country.
While the CNN/ORC International survey also indicated that slightly more people were angry at Republicans than Democrats or Obama for the shutdown, both sides were taking a hit.
According to the poll conducted over the weekend, 63% of respondents said they were angry at the Republicans for the way they have handled the shutdown, while 57% expressed anger at Democrats and 53% at Obama.
"It looks like there is more than enough blame to go around and both parties are being hurt by the shutdown," said CNN Polling Director Keating Holland.
Meanwhile, Treasury Secretary Jack Lew said on CNN's State of the Union that the government risks more than its credit rating if the debt ceiling is not increased by October 17. He dismissed suggestions that the government could avoid default by making only interest payments, saying Social Security payments and veteran's benefits could be endangered.
"It's very dangerous, it's reckless," Lew said.
If Congress fails to raise the debt ceiling, borrowing money to meet the nation's obligations won't be possible, CNNMoney's Jeanne Sahadi reported Monday.
Instead, Sahadi reported, lawmakers would have a few options to choose from that would have to be implemented right away -- cut government spending for the military and other discretionary programs by up to 33% every month; cut mandatory spending such as entitlement programs by 16% every month, and raising taxes by up to 12% every month.