The Sunday Herald has been told the Finance Secretary privately believes the Yes side could win by as much as 60-40.
Is he really making this adventurous prediction? "I think Yes can win it decisively," he says in the Melville Room at Glasgow University. "It would be good to have a clear, decisive result."
He explains the basis of his bold thinking: "I'm going out with people in my locality who I have never campaigned with in my life in substantial numbers.
"I think one of the key things Yes Scotland has done has been to encourage the establishment of, essentially, campaign teams beyond the SNP. It has been an immensely successful outreach campaign."
But Yes is behind in the polls and it seems time is running out for any sort of victory.
"There's a big undecided factor that's still in the polls," he says. "A large proportion of the electorate are just beginning to lock on. There's a lot of people out there who are just ready to listen to the arguments."
Swinney now is a very different figure compared to 10 years ago. In July 2004, he had just resigned the SNP leadership after being dogged by internal criticism.
He took over from Alex Salmond as leader in 2000 but quit after four years of in-fighting.
When he sees the First Minister leading the SNP into the referendum, does he ever think it could have been him?
"Not in the slightest. I gave my leadership of the SNP my very best. It wasn't my reserve strategy. It wasn't me putting it off to put out my best stall at a later stage. I gave it my best."
He adds: "He [Salmond] has taken the SNP and the argument further than I was able to do. That's said with an honest reflection."
However, Swinney, who turned 50 in April, paved the way for Salmond's success.
Under his leadership, the SNP backed one-member-one-vote for internal selections and modernised its campaigning structures.
"I feel as if I have helped along the way, [that is] how I would characterise it," he says. "As I look back on my term of leadership, the reforms that I undertook undoubtedly helped us to get us in fit shape for 2007."
Salmond won't be First Minister forever and speculation mounts about successors.
Deputy First Minister Nicola Sturgeon is the favourite, but some party insiders think Swinney could reclaim the top job.
But Swinney does not so much close the door on his leadership as slam and lock it.
Would you stand again for the SNP leadership? 'Unreservedly, absolutely, in no circumstances, there is no way of you configuring any signal I am possibly giving you, to say anything other than 'no.'"
He adds: "I came to a conclusion, a very tough personal conclusion, that it wasn't in the interests of the SNP for me to sustain that [his leadership]. I have certainly no intention of going back to that territory."
Does he think Sturgeon has got what it take to be First Minister? "Yes."
Swinney is so secure in his Cabinet role he can poke fun at his boss. Asked about the First Minister's diet, he said: "In the words of a political slogan: there's much that's been achieved, and much more to be achieved."
But some SNP sources say that while Swinney is an effective Finance Minister in a devolved context, he would not be the best choice for the same post in the event of independence.
This is because, they argue, he does not have the fire in his belly when it comes to redistribution and reducing inequality.
He replies: "I used to view issues of inequality as fundamentally issues of a social character, as being about the social inequity within our society, and the solutions lay in that territory.
"I have changed my thinking on that. Inequality, in whatever shape or form, eventually percolates into economic inequality. As a country, we can't afford economic inequality."
Asked whether he would like the rich to pay a higher rate of income tax after independence, he says: "There is no need for us to have higher taxes to pay for independence ... You then go onto a question about what's the right level of taxation. I've been very clear that the UK Government was wrong to remove the 50% tax rate when they did."
The First Minister has not supported the re-introduction of this policy, but does Swinney think there is a case for restoring the 50p rate in an independent Scotland?
"There is undoubtedly a case for that, of course there is," he says, adding that it not something his party has set out in its proposals.
Beyond being Finance Minister, Swinney is also MSP for Perthshire North.
What does he make of the late Tory MP Sir Nicholas Fairbairn, who used to represent a nearby seat, being linked to the Westminster child abuse scandal?
"It needs to be fully and comprehensively investigated," he says.
With time running out in the interview, I raise a remark Swinney made in 2003 which seemed to jar with his moderate political instincts.
Addressing the SNP conference as leader, he told activists to "tell the Brits to get off".
Does that phrase now make him cringe?
"That's probably the finest reflection of how I feel about that remark," he laughs. "It was a remark made at the end of a long day, and the end of a long process, and I wished I had never said it."
Enoch Powell famously said that all political lives end in failure, but Swinney's career proves that initial disappointment can lead to something more satisfying.