Two senior senators have taken on the fraught process of finding a congressional deal to reopen the US government and raise the country's borrowing limit, just four days before the Treasury is scheduled to run out of money.

The Democratic Senate majority leader, Harry Reid, and his Republican counterpart, Mitch McConnell, long-time adversaries, met on Saturday morning after it became clear that a separate attempt to broker a deal, between the Republican-dominated House of Representatives and the White House, had failed.

Focus shifted to the Senate when President Barack Obama rejected a plan from House Republicans to extend the so-called debt ceiling by just six weeks and made no offer lift the US government shutdown. Reid told reporters his meeting with McConnell had been "cordial" and added that they were only at preliminary stages.

"We don't have anything done yet – there's a long way to go before something like that will happen," he said, adding that he hoped to achieve a deal "in the next 48 hours".

Reid, McConnell and two other senators – the Tennessee Republican Lamar Alexander and New York Democrat Charles Schumer – met in Reid's office at 9am on Saturday, for just under an hour. Later in the day, Reid and Schumer met Obama in the Oval Office at the White House.

Charles Schumer. Photograph: EPA

The four senators were described by a senior congressional aide as a "nucleus" that could now rescue the country from default. However at a press conference, Reid and Schumer expressed only cautious optimism, and after a strategy meeting with fellow Democrats, the California senator Dianne Feinstein said she could not see any way through the current impasse. "There are people that have different proposals – there are probably four or five of those," she said. "But the proposals that will get the leadership of the House, the Senate, and the president, are not out there at this time."

On the Senate floor, a Democratic procedural measure that would have moved the Senate to a vote to extend the debt ceiling through the end of 2014, with no contingencies attached, fell after it failed to secure the 60 votes required.

There was then hope that a bipartisan bill, by the Maine Republican senator Susan Collins with input from a West Virginia Democrat, Joe Manchin, might offer a way forward. That would have increased the borrowing limit until January 2014 and reopening the government until March, in return for concessions aimed at pleasing Republicans, who precipiated the crisis when they failed to pass a budget resolution to ontinue the funding of federal services without measures designed to undermine the Affordable Care Act, Obama's signature healthcare reform.

However in a sign of the disarray that has affected an apparently directionless Capitol Hill, interest in that proposal lasted no more than a few hours.

Furloughed federal employees attend a protest outside the US Capitol to demand an end of federal workers' lockout after the shutdown. Photograph: Xinhua /Landov / Barcroft Media

The risk of a possible US default – which most economists agree would be catastrophic and reverberate across the world – has concentrated minds on Capitol Hill. Some Repulbicans have become increasligly concerned that they are being blamed for the crisis. They hoped that by focusing on the unpopularity of Obamacare and characterising the president's stance as a refusal to negotiate, that the White House would be seen as responsible for the first shutdown since 1996.

But an NBC News/Wall Street Journal poll this week showed the public largely blaming Republicans, with the party suffering its all-time worst rating since the survey began. Support for Obama's health reforms, which were introduced on the day the shutdown began, has increased despite early problems with enrolment websites.

Procedural complications

There appears to be little prospect of a deal before markets opened on Monday. All eyes are now on a deadline of Thursday when, according to the Treasury, the $16.7tn debt limit must be raised in order for the US to continue paying its creditors. Even if a deal between Reid and McConnell is reached over the next 24 hours, procedural complications mean a vote might not be taken until Wednesday or even Thursday, a senior congressional aide said.

Republican members of the House gathered for a private meeting with the speaker, John Boehner, on Saturday morning. The Republican leader informed his fellow representatives that discussions with the president had not yielded any prospect of a deal. "The president rejected our deal," said the Idaho representative Raul Labrador, immediately after the meeting. "It is now up to the Senate Republicans to hold firm." Representative Thomas Massey, of Kentucky, added: "We made an offer to the president and he turned it down. He's still refusing to negotiate – the president is refusing to offer anything."

House Speaker John Boehner leaves a Republican conference in the US Capitol on Saturday. Photograph: Jim Lo Scalzo/EPA

Obama has repeatedly said he will not be taken "hostage" by House Republicans, and will only enter into formal negotiations once they remove the treat of a US default and reopen the government. Speaking in his weekly address, on Saturday morning, Obama said he was against any stop-gap measures which would buy time but still leave the country under the shadow of a possible default.

"It wouldn't be wise, as some suggest, to just kick the debt ceiling can down the road for a couple months, and flirt with a first-ever intentional default right in the middle of the holiday shopping season," he said.

Even if a deal is reached between Reid and McConnell, with backing from the White House, the challenge will lie in securing the support of the Republican-dominated House of Representatives. Republican leaders now appear to have all but given up their initial efforts to defund Obamacare, or delay the "individual mandate" of Obamacare, which compels US citizens to obtain healthcare or face fines.

They now appear to be seeking some kind of concession that will justify a course of action that resulted in an initial 800,000 government workers being furloughed, or temporarily suspended from work, and damaged local economies across the country which are dependent upon federal funds.

The shutdown, now entering its 13th day, has laid bare the bitter divisions in the Republican Party. What began as classic congressional deadlock between a Republican-dominated House and a Democratic-controlled Senate has become an internecine dispute between the Tea Party movement and more moderate Republicans.

The Statue of Liberty will reopen to the public after the state of New York agreed to shoulder the costs of running the site during the federal government shutdown. Photograph: John Minchillo/AP

Its impact has been felt most acutely in Washington, where there is a concentration of employees of federal agencies and departments. At least 500,000 government workers are still thought to be furloughed. But across America federal outposts have also felt the squeeze, particularly in the many rural, local economies dependant upon national parks. One of the worst affected area is in the rural North Carolina district which elected Tea Party Republican congressman Mark Meadows.

He was described as the "architect" of the shutdown after orchestrating a petition among House Republican members to force the leadership to use the federal budget as leverage to undermine Obamacare.

Meadows denied he was particularly instrumental in starting the shutdown, telling the Guardian that he was striving to reopen the government for his constituents, many of whom are reliant on tourism from nearby national parks. "Honestly, if they're angry with me, I'm doing all I can to get the government back open and make sure the harmful affects don't hurt people back home."

Under a deal agreed reached late on Friday, the Statue of Liberty, Grand Canyon, Mount Rushmore as well as some national parks in Colorado and Utah reopened this weekend, after the government allowed them to be temporarily funded by state money and other non-federal sources.