Tomorrow's meeting between European Commission President Ursula von der Leyen and US President Donald Trump in Scotland is being treated as a "final negotiating session".
A senior official in the EU delegation told RTÉ News that there is "optimism but no certainty" that a trade deal can be reached.
The meeting is being treated as a "president-to-president" negotiating session.
The sentiment was echoed by a senior US source, who told Reuters: "We're cautiously optimistic that there will be a deal reached. But it's not over till it's over".
It's understood Ms von der Leyen, who set off for Scotland this evening, will also be joined by the European Trade Commissioner, Maroš Šefčovič, in Turnberry.
In the event that a deal is reached, it will be an agreement in principle, with a final document likely to be produced in the days afterwards.
Mr Trump said it would be the biggest trade agreement reached yet by his administration, surpassing the $550 billion accord reached with Japan earlier this week.
The meeting comes ahead of next Friday's deadline to reach a deal to avoid paying a steeper tariff rate.
Mr Trump had signed an executive order last month delaying an initial 9 July deadline to reach a deal.
Based on what is known so far, the EU-US deal would include a baseline tariff of 15%.
Some existing tariffs would be folded into that 15% meaning it would be an all-in tariff.
Mr Trump spent much of his day today playing golf at his Turnberry resort.
He is also British Prime Minister Keir Starmer and Scottish First Minister John Swinney on Monday.
Meanwhile, protests against the visit of Mr Trump to Scotland took place this afternoon.
Organised by the "Stop Trump Coalition", the demonstrations were held simultaneously in Edinburgh and Aberdeen.
Those involved said that they wanted to convey the message that they did not believe Trump should be welcomed to Scotland.
Tight security around the Trump Turnberry course meant no demonstrators were seen when the president took to the greens this morning.
Dressed in black, with a white cap that said USA on it, Mr Trump could be seen driving a golf buggy, flanked by a fleet of security personnel, as he played on the famous course which he bought in 2014.
However hundreds gathered in both Edinburgh and Aberdeen - near the site of Mr Trump's other Scottish golf resort - to make their opposition to the president known.
As the visit got under way Scottish First Minister John Swinney announced public money to support a tournament at the Trump International golf links in Aberdeenshire.
The 2025 Nexo Championship - previously known as the Scottish Championship - is set to take place there next month, supported by £180,000 of public funding.
Mr Swinney said: "The Scottish Government recognises the importance and benefits of golf and golf events, including boosting tourism and our economy."
But Scottish Green co-leader Patrick Harvie likened the awarding of public cash to the tournament to "handing some pocket money to the school bully".
Mr Trump will head to his golf resort in Balmedie, Aberdeenshire later on in his five-day long private visit.
As part of his trip he will also hold talks with Prime Minister Keir Starmer, where the two men will refine a trade deal between the UK and the US that was agreed earlier this year.
The president is also expected to talk trade with European Commission president Ursula von der Leyen tomorrow.
But with no talks apparently scheduled for today, the president, who is well known for his love of golf, was free to take to the famous course at Turnberry.
A major security operation surrounded him, with police officers and military personnel seen scouring the grounds ahead of Mr Trump teeing off.
After touching down in Scotland in Air Force One last night, the president was seen on the course at Turnberry this morning - although security fencing had also been erected around the resort, with road closures also in place to help protect Mr Trump, who last year survived an assassination attempt.
Police Scotland has asked for support from other forces across the UK to bolster officer numbers for the high-profile visit - with Mr Swinney appealing to people to protest "peacefully and within the law".
In Aberdeen, Green north east Scotland MSP Maggie Chapman told the crowd of hundreds: "We stand in solidarity not only against Trump but against everything he and his politics stand for."
Speaking about the US president, Ms Chapman said: "He believes that climate change isn't real, he believes that cutting services for those in the world with the least is the right thing to do.
"We say no to all of those things, not in our name, never in our name."
With Mr Trump having last year been convicted by a New York jury of falsifying business records, she said the president was a "convicted felon".
"He is not welcome in Scotland, he is certainly not welcome in Aberdeenshire," she said.
"We know that he is a convicted felon.
"We also know that all of the promises he has repeatedly made to Scotland have come to nothing, there hasn't been the development of jobs or houses that he promised when he opened his course in Aberdeenshire a few years ago."
Mr Harvie was also critical of the president, saying: "He's attacking our renewables industry once again, one of the success stories of Scotland, and he is trying to attack and undermine it.
"And that is after having trashed part of our natural environment on the Aberdeenshire coast to build his golf resort."
