The main principles of a compromise emerged in public statements from both sides.
It will include an extension - temporary or permanent - to US borrowing authority, without which Washington could begin to default on its obligations for the first time in history after Thursday.
The Government, shutdown since October 1, will be fully re-opened, possibly on an interim basis, and there will be some kind of commitment from both sides to work towards an elusive deal to tackle the deficit, rein in spending and possibly to reform social programmes and some aspects of the tax code.
Obama and Boehner, longtime foes who have often struggled to close deals, had what both sides said was a constructive conversation.
But there were signs that the White House was driving a hard bargain, as Carney said a rise in the debt ceiling could not be linked to long-term fiscal talks with Republicans, because it could set up repeated threats of default in the coming months.
The White House said earlier in the week that it would be open to a six-week extension of the debt ceiling after the Thursday deadline.
It now appears to be looking for a longer-term extension of borrowing authority from the current US$16.7 trillion ($20.1 trillion) level.
One possible compromise plan is being offered by Republican Senator Susan Collins.
Her measure would raise borrowing authority for up to a year, fund the government - and repeal a tax on medical devices introduced under Obama's health care law, giving conservative Republicans an incentive to back it.
Such a solution would be a facesaver for Republicans who entered the showdown determined to use the need to raise the debt ceiling as a way to de-fund or delay Obamacare.
Republicans from the lower house have offered to talk to Obama about a short-term resolution to fund the Government, and then move to long term budget and fiscal talks.
But the President insists he will not negotiate until the debt ceiling is raised and the Government is re-opened.
Republican senators emerged from talks with the president at the White House expressing optimism.
Republican Senator Lindsey Graham said he was "confident that in the next 24 to 48 hours the House will produce a continuing resolution that will allow the Government to be open in total".
The Senate will today vote on a 15-month extension of the debt ceiling, to take the issue off the table until after the mid-term congressional elections next year.
While the measure is unlikely to become law because it needs House support, it could provide a template for an eventual solution in the event of a longer term fiscal deal between Obama and Republicans.
Hopes for a deal spurred a rally in Asian stocks and Wall St followed the trend into positive territory in early trade.
If the US debt ceiling is not raised by Thursday, the Treasury would run out of money and could begin defaulting on US obligations for the first time in history, with probable dire consequences for the world economy.
New signs of Republican flexibility may have been spurred by devastating polling numbers showing their party has been badly hurt by the political showdown.
An NBC News/Wall Street Journal poll showed that 53 per cent of those asked blamed Republicans for the government shutdown and only 31 per cent pinned the blame on Obama.
And 70 per cent of those polled blamed Republicans for putting their political agenda before what was good for the country, compared to 51 per cent who said the same of Obama.
G20 finance ministers meeting in Washington meanwhile demanded urgent action.
- AAP
