A flurry of meetings were held throughout the day, with Unite saying it was working to persuade owners Ineos to reverse its shock decision to close the petrochemical complex.
General secretary Len McCluskey said the union had decided they had to embrace the survival plan, "warts and all", in the wake of the closure decision.
Managers were said to be discussing "everything that has been said", before deciding whether to reverse the decision to close the petrochemical side of its business, with the loss of 800 direct jobs and up to 2,000 contractors.
An announcement from the company is expected tomorrow.
Meanwhile, UK government officials admitted it would be a "challenge" to find another company to buy the business, which Ineos has said is losing £10 million a month.
The losses, scale of investment needed to upgrade the site, coupled with the industrial relations would all have to be considered by any potential buyer, officials said.
But they insisted there would be no shortage of fuel supplies as a result of the dispute, which has led to the petrochemical site and adjoining oil refinery being closed for the past week.
"We don't envisage there will be shortages," said one official.
Ministers from the Scottish and UK government held talks with Unite and management today, saying later that both administrations were doing all they can to keep the site open.
Workers at the petrochemical site, and adjoining oil refinery, had refused to sign up to a survival plan, which included a pay freeze, ending of the final salary pension scheme, and other changes to terms and conditions.
Mr McCluskey said: "We are not going to let this plant close. We are encouraged by the comments of the First Minister that he too will not let this plant close.
"We have a situation whereby a company has put down an ultimatum and we have to respond. It is not how we engage in modern day industrial relations.
"My union is engaged with thousands of companies every day to negotiate plans to save jobs. There is nothing humiliating about negotiating plans to ensure jobs and communities are safe.
"This plant is on cold shut down and each day that goes by makes it harder to start back up again, which is why the stewards made the offer to the company - so that we can get people back to work."
The company insisted yesterday it had no alternative but to close the business after it failed to persuade its staff to accept a survival plan, saying that white-collar workers such as admin staff had backed the plan, but workers represented by Unite had rejected it.
Politicians have urged the two sides to resume talks to prevent the closure, while efforts are expected to be made to find a potential buyer.
Tom Crotty, a director of Ineos, said the company would put Unite's proposals to shareholders if there was a "very significant" change from the union.
He said: "The management team on the site will listen to hear what Unite has said.
"If they believe there are substantive differences in where we are now then clearly they have a duty to take that back to the shareholders.
"The shareholders met after the vote on Monday and, quite understandably in my opinion, took a view that if the workforce had rejected that £300 million investment, then how were they going to go ahead and make it if the workforce were not behind that? That is why the closure announcement has been made.
"If the management team feel that there is a very significant change then I'm sure they would probably feel they would need to take that back to the shareholders and have further discussions."
Finance Secretary John Swinney said after today's meeting that the company had to "absorb" what Unite had offered, adding that the two sides had to come together to resolve the dispute.
He said the plant had a "great future".
Scottish Secretary Alistair Carmichael said there was a better prospect of an agreement following the shift by Unite.
Mr McCluskey has told the company he wants to meet owner Jim Ratcliffe for face-to-face discussions.
Prime Minister David Cameron said he was keen for a resolution between both sides.
"I welcome the fact that the trade unions and the business are working together to seek a solution," he said.
"This is an important business for Scotland, it's a very important industry for the whole of the United Kingdom. We want to see those jobs saved, we want to see this business thrive, and I'm hopeful that agreements will be reached."
Labour leader Ed Miliband said he had discussed the crisis with Unite.
"I have spoken to them, and I think they fully understand the gravity of this situation.
"They want to represent their members, and rightly so, and they're concerned about their members' jobs and livelihoods.
"I have a concern about that but also making sure that we have proper energy supplies for this country. That's why it's a national asset."
He added: "I think we've seen very significant movement on the union side, rightly calling off the industrial action - the strike - and now looking like they're open to changes on the terms and conditions.
Energy and Climate Change Secretary Ed Davey said: "Government wants to see a long-term future for the 800 jobs at the petrochemical plant, which is so crucial for the workforce, the communities the plant supports and the wider Scottish economy.
"As we make progress towards a resolution, I now urge both sides to work together in good faith to ensure that this long-term future is secured."
Following talks at the plant Mr Carmichael said: "It's clear that we're dealing with a different situation today following the statement from Unite that they were prepared to accept the Ineos survival plan without any pre-conditions.
"I think it's fair to say that we are in a much better place today in relation to the future of the plan than we were yesterday. There remains of course a great deal to be done."
Mr Swinney said: "The management were open with us about the details of the conversation they've had with the trade union and the trade union were equally clear with me that there has been a complete acceptance by the trade union of the management survival plan."
Mr Carmichael said the decision about the future of the plant lies ultimately with company shareholders.
He said: "I think you have to be careful before you start getting too carried away. There's still a great deal of ground to be covered.
"What I detect as a result of the changed union position is a willingness to start covering that ground."
First Minister Alex Salmond began Grangemouth talks with Mr McCluskey and Unite's Scottish secretary Pat Rafferty at around 4.30pm in Edinburgh today.
The three men made no comments before entering the meeting at the Scottish Parliament.
A briefing paper by government officials said UK refineries operated in a "challenging" international market for crude oil and refined products.
Margins are being "squeezed" amid overcapacity in global refining.
The UK is becoming more reliant on imports of diesel and aviation fuels, as refineries in this country were mainly built to meet demand for gasoline.
"Put simply, UK refinery output is out of balance with market demand, and profit margins are thin.
"Competition from the US, with cost advantages due to shale oil and gas, is expected to exacerbate this situation," said the paper.
Officials pointed out there were no shortages when the Coryton refinery in Essex closed last year.
Contingency plans already in place include increasing imports as well as output at other terminals, including Ross and Clydebank.
Hauliers also have "strong" arrangements in place, including provision for long range deliveries from the North of England.