Meanwhile, the UK's two largest oil companies, Shell and BP, have delivered their latest financial results amid record petrol prices at the pumps.
Shell reported first quarter profits of $7.7bn (£3.92bn) for the three months to 31 March, while BP posted $6.59bn (£3.32bn) first quarter figures, a rise of 48% from $4.4bn.
The Grangemouth strike, which began on Sunday, halted almost half the UK's North Sea oil production when a key pipeline powered by the plant was closed.
BBC Scotland correspondent Colin Blane said it would have cost the economy £50m a day.
He said the unions were considering whether to give notice of another possible strike as the deadlock over pensions had not been broken.
Mr Hutton and Mr Swinney, however, are encouraging the two sides to work together to ensure production at the refinery is back up to capacity as quickly as possible.
Mr Hutton is also set to hold talks with representatives from the oil and gas industries as well as members of the Scottish Government.
The two-day walkout came amid a row over pensions
Speaking at Grangemouth, Mr Hutton said: "I'm glad that the union said that there should be a pause for reflection and I'm very glad that later today there will be discussions between the unions and the company to try and find a way of resolving this dispute.
"There is a gap between the two sides that has got to be bridged - only the two parties themselves can reach an agreement.
"No-one can do that for them. But I hope there is now a mood to try and reach an agreement."
Mr Swinney said it was now incumbent on those at the centre of the dispute to work together.
He added: "What the two governments will be encouraging is a process of dispute resolution between the two parties, that they focus on those essential differences and work to resolve them and to avoid any possibility of further strike action."
The Scottish Government is shipping about 65,000 tonnes of fuel, mostly diesel, from Europe to bolster supplies.
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