Republicans in the US House of Representatives and President Barack Obama have met amid efforts to avert a looming debt crisis.
Both sides said the 90-minute discussion was useful but no decision was made. They agreed to keep talking.
Republicans have offered the president a short-term debt limit increase to stave off default.
A new survey suggests the majority of Americans blame the Republicans for the partial shutdown of government.
A Wall Street Journal/NBC News poll suggests 53% of Americans fault Republicans for the crisis, compared with 31% who say the Democrats are responsible.
"It was a very adult conversation," said Republican Hal Rogers of Thursday afternoon's meeting. "Both sides said they were there in good faith."
House Majority leader Eric Cantor called the meeting "very useful" and said the talks were continuing.
Mr Obama held a White House meeting on Friday with Republican senators, who said they would present options of their own for ending the budget standoff.
Officials have warned the US risks default on 17 October if the nation's $16.7 trillion (£10.5 trillion) borrowing limit is not increased.
Republicans have offered to extend the government's borrowing authority beyond the deadline, temporarily putting off a default.
In return they want Mr Obama to negotiate on the budget dispute that has partially closed the government - the first such shutdown for 17 years.
Earlier, House of Representatives Speaker John Boehner said: "It's time for leadership. It's time for these negotiations and this conversation to begin."
A spokesman for Mr Boehner told reporters the deal offered was a "clean" increase of the debt limit, with no additional policies attached, lasting six weeks - until 22 November.
But it is not clear if Republicans are willing to drop entirely their attempts to defund or delay Mr Obama's 2010 healthcare law.
Reacting to the offer, White House press secretary Jay Carney told a daily press briefing the president was glad that "cooler heads" seemed to be prevailing in the House.
But he added: "He will not pay ransom in exchange for the Republicans in the House doing their job."
Earlier on Thursday, Democratic Senate leader Harry Reid said the Senate would "look at anything" the House sent to them, but they would not engage in negotiations with Republicans prior to reopening the government.
Rattled US stock markets rebounded on Thursday on news of a possible breakthrough.
Hundreds of thousands of federal employees have been out of work since the shutdown began, and private firms, from arms makers to motels, have begun to lay off workers.
About 15,000 private-sector employees have filed for unemployment benefits due to the shutdown, the US labour department said on Thursday.
And governors in at least four western states - Utah, South Dakota, Arizona and Colorado - have asked for authority to reopen national parks within their borders because of the economic impact of the closures.
On Thursday, Mr Obama signed legislation restoring death benefits to families of US troops who have died since the government closed. The shutdown prevented processing of the payments, typically made within days of the soldier's death.
The president met House Democrats at the White House on Wednesday and told them he would prefer a longer-term increase to the nation's debt ceiling, but was willing to accept a short-term fix.
Earlier, Treasury Secretary Jack Lew told a Senate hearing that skipping a payment on US debt would trigger financial crisis"chaos".
Since the US hit its debt ceiling in May, the treasury has been using what are called extraordinary measures to keep paying the bills, but those methods will be exhausted by 17 October, Mr Lew has said.